Tensor Signal Processing

Turning sparse wireless measurements into structured maps for spectrum sensing, radio mapping, source localization, and MIMO beam-aware communications.

Why tensors?

A tensor is a multi-dimensional array. A vector is a 1D list, a matrix is a 2D table, and a tensor is what we use when the data has three or more dimensions. In wireless systems, this happens all the time: signal power may vary with horizontal position, vertical position, frequency, time, antenna beam, or propagation path. Instead of flattening everything into one long table, a tensor keeps these relationships visible. This is useful because wireless data is not random pixel noise: nearby locations, nearby frequencies, and nearby beams often share hidden structure.

Two examples from our recent work are shown below. In spectrum-map construction, we model a 3D tensor whose slices are power maps at different frequency bands:

H = \sum_r S_r \circ \phi_r .

Here, \(H\) is a 3D spectrum-map tensor, \(S_r\) is the 2D spatial propagation map of source \(r\), \(\phi_r\) is the 1D frequency signature of that source, and \(\circ\) means an outer product. The advantage is that we do not reconstruct each frequency map independently. If one transmitter is blocked at one location, its power is often weakened across several frequency bands, and the tensor model shares this information across the whole cube.

In beam-map reconstruction in a 5G MIMO system (a system with many antennas), we use a related tensor model after transforming positions from Cartesian coordinates to polar coordinates:

X = \sum_r G_r \circ \rho_r .

Here, \(X\) is a 3D beam-map tensor, \(G_r\) describes how transmit beam direction and user angle interact, and \(\rho_r\) describes how the signal decays with distance. The key insight is that the beam-angle matrix often has a diagonal-stripe, Toeplitz-like pattern caused by the shift-invariant structure of antenna-array responses. This turns a difficult map-reconstruction problem into a structured signal-processing problem.

Two tensor models at a glance

Figure 1. Spectrum-map tensor model. Sparse measurements are converted into a completed 3D spectrum map by combining interpolation with block-term tensor decomposition. Each source contributes a spatial propagation map and a binned frequency spectrum.

Toeplitz-structured MIMO beam-map tensor model

Figure 2. MIMO beam-map tensor model. A wireless scenario is represented as a tensor, which is decomposed into a Toeplitz beam-space structure and a distance attenuation vector.

A typical workflow: spectrum map construction

Let us focus on one application: constructing a power spectrum map from sparse measurements. Imagine that a few sensors, vehicles, or UAVs move through an area and record received signal power at several frequency bands. The measurements are useful, but they are usually incomplete: sensors are not placed exactly on a neat grid, different sensors may scan different frequency bands, and many locations are never measured.

Step 1: Organize the data as a tensor

We first define a grid over the target area and stack the frequency bands as the third dimension. Each slice of the tensor is a 2D power map at one frequency band. The observed entries come from the sparse measurements, and the missing entries are the places and frequencies that we want to predict.

Step 2: Couple interpolation with tensor decomposition

A standard interpolation method estimates missing values using nearby sensors. This is helpful, but it only sees local neighborhoods. A tensor model adds global vision: it learns that the same source produces related spatial patterns across many frequency bands. However, a conventional tensor completion method easily runs into an identifiability issue for insufficient measurements. Our integrated interpolation and block-term tensor decomposition method lets the local interpolation and the global tensor structure help each other instead of running them as two disconnected steps. Such integrated approach naturally resolves the identifiability issue.

Step 3: Use the reconstructed tensor

The completed tensor can be used to visualize spectrum occupancy, separate and localize multiple transmitters, identify strong interference regions, guide new sensor placement, and support radio-resource decisions. In practice, the map can also be updated as new measurements arrive, which makes it suitable for sensing and optimization in dynamic wireless networks.

Theoretical Foundations: Low-Rank Geometry, Error Scaling, and Identifiability

References [1]-[4] in this section refer to the works listed under Ours in the suggested reading.

A central theme in our work is that wireless propagation maps are not arbitrary images. They contain mathematical structures induced by physics: signal strength usually decays with distance, nearby frequencies share propagation patterns, antenna arrays create shift-invariant beam responses, and spatial maps often have low-dimensional representations. Our research tries to make these structures explicit and then design algorithms around them.

1. A propagation field has a rank-one backbone

Consider the propagation map generated by one source at location \(s\). After discretizing the area into a matrix, we may write

H^{(s)}_{ij}=g(\|c_{ij}-s\|),

where \(c_{ij}\) is the center of grid cell \((i,j)\), and \(g(d)\) is a non-negative decreasing function of distance. A key theoretical discovery is that the dominant left and right singular vectors of \(H^{(s)}\) are unimodal: each vector increases toward the source coordinate and then decreases away from it.

In other words, if

H^{(s)} \approx \sigma_1 u_1 v_1^\top ,

then the peak of \(u_1\) indicates one coordinate of the source, and the peak of \(v_1\) indicates the other coordinate. This result is important because it says that a rank-one approximation captures the main geometry of a single-source propagation field, even when the exact path-loss function \(g(d)\) is unknown.

This also gives a theoretical explanation for why matrix and tensor methods can localize sources from sparse measurements. Under regularity conditions, the mean squared localization error can decay faster than

O(M^{-1.5}),

where \(M\) is the number of measurements. By comparison, a naive peak detector or weighted centroid localization method typically has a slower \(O(M^{-1})\)-type scaling. Thus, the low-rank structure is not only visually intuitive; it leads to a provable improvement in estimation accuracy.

2. Unimodality helps separate multiple sources

For multiple sources, the observed propagation map becomes a mixture of several source fields. The dominant singular vector of the whole matrix may no longer correspond to one source. To address this, we introduced unimodality-constrained matrix factorization, where the propagation matrix is decomposed into several unimodal components:

H \approx U V^\top .

The columns of \(U\) and \(V\) are constrained to be unimodal. Each pair of unimodal vectors corresponds to one source.

A further theoretical insight is that the coordinate system itself matters. For two sources, one can rotate the coordinate system and examine the normalized dominant singular value

\rho(\theta)=\frac{\sigma_1^2(H(\theta))}{\|H(\theta)\|_F^2},

where \(H(\theta)\) is the observation matrix constructed after rotating the coordinate system by angle \(\theta\). When the two sources become aligned with one axis, the matrix becomes closer to rank one, and \(\rho(\theta)\) is maximized. Under mild conditions, this function has a locally unimodal structure, which makes efficient rotation search possible. This result shows that matrix structure, source geometry, and algorithm design are tightly connected.

3. Interpolation error follows a bias–variance law

Tensor and matrix completion methods are powerful, but in practice the measurements are often off-grid: sensors are not located exactly at the centers of the grid cells. This motivated our interpolation-assisted matrix completion framework.

Let

\xi(c)=\hat{\rho}(c)-\rho(c)

be the interpolation error at location \(c\), where \(\hat{\rho}(c)\) is the local polynomial estimate of the propagation field. If the interpolation window has radius \(b\), then the theory shows a clear bias–variance tradeoff:

\text{Bias}(\xi)=O(b^2), \qquad \text{Var}(\xi)=O\!\left(\frac{1}{M b^2 f(c)}\right),

where \(M\) is the number of measurements and \(f(c)\) is the local sensor density around \(c\). A large window reduces noise because it uses more measurements, but it increases bias because the local polynomial model becomes less accurate over a larger region. A small window has less bias but higher variance.

Balancing these two terms gives an optimal scaling

b \sim M^{-1/6},

and the interpolation mean squared error scales as \(O(M^{-2/3})\). This result provides a principled way to choose the interpolation window, rather than tuning it by trial and error.

4. First-order interpolation removes sensor-density bias

The interpolation analysis also reveals why a first-order local polynomial model can be better than a simple weighted average. For zeroth-order interpolation, the bias contains a term related to the sensor-density gradient:

\nabla f(c)\cdot \nabla \rho(c).

This means that if sensors are unevenly distributed, a simple local average may be biased toward the denser side of the measurement region. First-order interpolation removes this term, leaving the bias mainly controlled by the curvature of the propagation field. This explains why first-order interpolation becomes more effective when enough measurements are available.

5. Uncertainty-aware completion connects local and global structure

The interpolation theory gives not only an estimate \(\hat{H}_{ij}\), but also an uncertainty level. If the interpolation error at grid cell \((i,j)\) has approximate mean \(\mu_{ij}\) and standard deviation \(\nu_{ij}\), then matrix completion can be made uncertainty-aware. For example, a trust region can be written as

H_{ij}-\hat{H}_{ij} \in \mu_{ij}\pm z\nu_{ij},

where \(z\) is chosen according to the desired confidence level. Equivalently, entries with smaller uncertainty can be assigned larger weights in a weighted matrix factorization problem.

This is the key idea behind interpolation-assisted matrix completion: local interpolation provides dense but uncertain observations, while low-rank completion uses global structure to regularize them. The two parts are not simply placed in a pipeline; the uncertainty analysis tells the global reconstruction algorithm how much each local estimate should be trusted.

6. Tensor structure improves spectrum-map interpolation

For spectrum maps, the data naturally forms a tensor because power varies across two spatial dimensions and frequency. The block-term tensor model, \(H=\sum_r S_r\circ \phi_r,\) says that different frequency slices are coupled through the same underlying source maps.

For a single source, our analysis shows that the tensor-guided interpolation error variance \(E_t\) is no larger than the error variance \(E_p\) of conventional frequency-by-frequency interpolation. More explicitly, the error depends on the source spectrum \(\phi\). A uniform wideband spectrum helps suppress frequency-selective fading at high signal-to-noise ratio, while a concentrated narrowband spectrum can be beneficial at low signal-to-noise ratio because it increases effective received power. Thus, the theory explains not only that tensor structure helps, but also when different spectral shapes make map construction easier or harder.

7. Identifiability depends on source diversity

Tensor decomposition can separate multiple sources because different sources usually have different spatial maps and different spectra. However, the theory also shows a limitation: if two sources have highly overlapping spectra, separation becomes more difficult. Let \(\eta\) denote the spectrum-overlap ratio. As \(\eta\) increases, the reconstruction error grows; when \(\eta\) approaches one, the two sources become nearly indistinguishable from power measurements alone.

This gives an identifiability message: tensor methods are powerful, but they still require diversity. The diversity may come from frequency, space, beam direction, time, or propagation path. Much of our research is about discovering and exploiting the right form of diversity for each wireless sensing problem.

8. MIMO beam maps have Toeplitz structure from array physics

For MIMO beam maps, the tensor structure comes from antenna-array responses. After transforming the map into polar coordinates, the beam-space matrix \(G\) has entries that depend mainly on the difference between transmit beam direction and user angle. Under uniform angular sampling in the sine-angle domain,

G_{ij}=G_{i+1,j+1}.

This is a Toeplitz structure: entries along the same diagonal are equal or similar. In a pure line-of-sight region, this reduces the number of unknowns in an \(N\times N\) beam-space matrix from \(N^2\) to roughly \(N\). This explains why Toeplitz-structured tensor decomposition can reconstruct MIMO beam maps from sparse beam measurements more efficiently than generic interpolation or generic tensor completion.

What performance advantage do we get?

The representative advantages in this section refer to Ours in the suggested reading.

The main benefit is data efficiency. Tensor structure lets the algorithm borrow information across frequencies, beams, angles, and distances. This means that fewer measurements are needed to obtain a useful map.

Scenario	What the tensor model uses	Reported advantage
Spectrum map with sparse sensors and multiple frequency bands	Shared spatial structure across frequency slices; source spectra separated by block-term tensor decomposition	Over 20% reconstruction accuracy improvement; roughly half the measurements needed for similar error in some settings
Off-grid measurements	Integrated local interpolation and global tensor structure	Stable reconstruction under off-grid measurements, with over 20% improvement over baselines
Multiple sources with overlapping spectra	Tensor factors separate spatial propagation maps and source frequency signatures	Better source separation; more than 10% improvement in reconstructed spectrum maps in representative tests
Propagation map reconstruction	Local polynomial interpolation plus uncertainty-aware low-rank matrix completion	10%–50% reconstruction MSE reduction over Kriging and other baselines in medium-to-large measurement regimes
RSS-based source localization	Dominant singular vectors of the reconstructed propagation map	More than 50% localization RMSE reduction compared with weighted centroid localization in representative experiments
MIMO beam map with sparse beam measurements	Polar-domain matrix-vector tensor decomposition and Toeplitz-like beam-space structure	Over 20% improvement in sparse LOS/reflection settings; over 40% in a LOS-plus-obstruction scenario

These values summarize representative numerical results from the cited works. They should be read as evidence of the value of tensor structure, not as a universal guarantee for every deployment.

Why This Research Matters

Tensor signal processing provides a bridge between mathematical theory and real wireless systems.

For theoretical research, this direction offers interpretable model-based tools that complement modern machine learning. Instead of treating radio maps as ordinary images, tensor models use physical structure: frequency correlation, spatial propagation, antenna-array geometry, blockage, reflection, and distance-dependent attenuation. This makes the methods easier to analyze, easier to diagnose, and often more data-efficient.

For real industrial applications, the value is practical: better maps with fewer measurements can reduce sensing cost, shorten drive tests or UAV flights, improve spectrum monitoring, support beam management, and enable environment-aware 6G networks. Tensor signal processing is therefore not only a theoretical topic; it is a tool for building wireless systems that understand the environment around them.

For prospective students who are interested in theoretical research, it offers a concrete way to learn optimization, statistical signal processing, low-rank modeling, and wireless communications through problems that can be measured, simulated, and deployed. One can start from a simple question: if we only measure a small part of the wireless environment, what structure lets us reconstruct the rest?