Tensor Decomposition

Expresses a tensor as a sum of a finite number of rank-one components. In RF fingerprinting, CPD is used to blindly separate mixed emitter signals by decomposing a cumulant tensor into distinct source signatures.

• Represents a tensor as a sum of outer products of vectors
• Essential for blind source separation of co-channel transmitters
• Uniqueness guaranteed under mild conditions without orthogonal constraints
• Directly extracts the signature vector for each individual emitter

Decomposes a tensor into a core tensor multiplied by factor matrices along each mode. This model provides a compressed, low-rank representation of a cumulant tensor while preserving multi-linear interactions.

• Also known as Higher-Order SVD (HOSVD)
• Yields a dense core tensor capturing interactions between components
• Factor matrices represent the principal features along each statistical mode
• Ideal for dimensionality reduction before classification

Represents a high-order tensor as a chain of low-dimensional 3-way core tensors (carriages). This format breaks the curse of dimensionality, enabling the storage and manipulation of very high-order cumulant tensors.

• Scales linearly with tensor order, not exponentially
• Each carriage connects to the next via a shared bond dimension (rank)
• Enables efficient computation of higher-order statistical moments
• Suitable for resource-constrained edge deployment

A relaxed variant of CPD that allows one factor matrix to vary across slices, making it robust to time-varying channel conditions. It is applied when emitter signatures drift or when signal alignment is imperfect.

• Handles shifts in one mode (e.g., time or frequency)
• Maintains uniqueness properties of standard CPD
• Robust to misaligned signal segments in dynamic environments
• Used for tracking emitters with drifting hardware impairments

Imposes non-negativity constraints on the CPD or Tucker model. In the RF domain, this ensures that extracted spectral components correspond to physically additive power contributions, enhancing interpretability.

• Decomposes tensors into additive, parts-based representations
• Factor matrices represent non-negative spectral bases
• Eliminates cancellation artifacts common in unconstrained models
• Produces physically meaningful power spectral components

Decomposes a tensor into a sum of low-multilinear-rank terms (blocks) rather than rank-one terms. BTD bridges the gap between CPD and Tucker, modeling emitter groups with shared internal structure.

• Each block has its own multilinear rank
• Models clusters of emitters with similar hardware impairments
• More flexible than CPD, more structured than Tucker
• Useful for hierarchical device identification in heterogeneous fleets

A multi-dimensional array organizing higher-order cumulants (typically third or fourth order) that serves as the primary input to tensor decomposition algorithms. Unlike a flat feature vector, the cumulant tensor preserves the multi-modal correlations between different time lags, frequency bins, or antenna elements.

• Stores joint statistical relationships across multiple dimensions
• Enables blind source separation without prior knowledge of mixing matrices
• Fourth-order tensors are theoretically blind to Gaussian noise, isolating non-Gaussian emitter signatures
• Decomposed via CP or Tucker models to extract rank-1 components representing individual sources

A multi-linear generalization of the standard SVD that decomposes a tensor into a core tensor multiplied by orthogonal factor matrices along each mode. HOSVD provides an optimal low-rank approximation in the Frobenius norm sense.

• Computes mode-n singular vectors by flattening the tensor along each dimension
• The core tensor captures interactions between components across modes
• Serves as initialization for more refined CP decomposition algorithms
• Enables dimensionality reduction by truncating singular vectors per mode independently
• Often used as a pre-processing step before classification to compress cumulant tensors while preserving discriminative structure

CANDECOMP/PARAFAC decomposition expresses a tensor as a sum of rank-one components—outer products of vectors. For RF fingerprinting, this factorizes a cumulant tensor into interpretable source signatures.

• Each rank-one component corresponds to an individual emitter or signal source
• The number of components (rank) must be specified or estimated via core consistency diagnostic
• Uniqueness is guaranteed under mild conditions, unlike matrix factorizations
• Alternating Least Squares (ALS) is the standard optimization algorithm
• Directly enables unsupervised emitter separation from mixed signal observations

An algebraic technique that simultaneously diagonalizes a set of cumulant matrices (slices of a tensor) to achieve blind source separation. Unlike iterative gradient methods, this approach yields closed-form solutions.

• Exploits the multi-linear structure of fourth-order cumulants
• Requires no step-size tuning or convergence monitoring
• Robust to local minima that plague optimization-based ICA
• Particularly effective when sources exhibit distinct non-Gaussian signatures
• Computationally efficient for moderate numbers of sources and sensors

A dimensionality reduction technique that projects signal data onto directions maximizing non-Gaussianity, isolating the subspace containing hardware-specific fingerprint information before tensor decomposition.

• Uses kurtosis or negentropy as projection indices
• Separates Gaussian noise subspace from non-Gaussian signal subspace
• Reduces tensor dimensions prior to decomposition, lowering computational burden
• Enhances signal-to-noise ratio for subsequent cumulant estimation
• Critical when emitter signatures are buried below the noise floor

An information-theoretic measure of irregularity in the bispectrum distribution that quantifies the complexity of non-linear signal interactions. When computed on decomposed tensor components, it provides a compact scalar feature for device discrimination.

• Measures the flatness or peakiness of the bispectral distribution
• High entropy indicates distributed, noise-like non-linearities
• Low entropy reveals concentrated phase coupling characteristic of specific hardware impairments
• Computed on individual rank-one components after CP decomposition
• Serves as a robust, rotation-invariant feature for emitter classification