Universal Three-Channel Decomposition

• Universal three-channel decomposition is a framework that isolates three distinct structural contributions by reducing large-scale problems into smaller, parameterized subproblems with analytic gradients.
• It employs methodologies such as constrained low-rank factorization, Tucker decompositions, and eigendecomposition to enable robust, efficient component extraction in diverse applications.
• The approach extends to quantum information and optical networks by leveraging symmetry, tensor unfolding, and identifiability conditions to ensure unique and interpretable decompositions.

Universal three-channel decomposition denotes a family of decomposition frameworks rather than a single formalism. In the cited arXiv literature, the expression and closely related constructions appear in constrained low-rank matrix approximation for three-way decomposition (Ibragimov et al., 2017), multivariate nonstationary signal separation with at least three sensors (Stankovic et al., 2019), Tucker-type reductions for 3D MIMO and beyond diagonal RIS channel models (Yuan, 2015, Almeida et al., 2024), convex and geometric decompositions in quantum information (Wang, 2015, Cho et al., 18 Sep 2025), permutation- and SU(3)-based decompositions of three-channel optical networks (Guise et al., 2014), and uniqueness theory for 3-tensor rank decompositions (Gubkin, 2021). In these works, “universal” refers to different kinds of scope: independence of particular overlap patterns, applicability to arbitrary antenna arrays, validity for arbitrary Petz monotone metrics, or decomposition statements for arbitrary channels within a specified class.

1. Terminological scope and general mathematical pattern

A common structural motif is the replacement of a large ambient problem by a smaller parametrized one. In the constrained matrix-factorization setting, the problem is to approximate $A \in {\mathbb C}^{m \times n}$ by $BC^*$, with $B \in {\mathbb C}^{m \times r}$, $C \in {\mathbb C}^{n \times r}$, while the entries of $B$ depend on a much smaller parameter set $\sigma_1,\dots,\sigma_k$, $k \ll mr$. For fixed $B$, the optimal factor is

$C = A^* B (B^* B)^{-1},$

and the minimization reduces to

$$\min_{\sigma_1,\dots,\sigma_k} \|A - B(B^*B)^{-1}B^*A\|_F,$$

or equivalently to

$BC^*$0

with $BC^*$1 an orthonormal basis for the columns of $BC^*$2 (Ibragimov et al., 2017). The paper states that this transformation yields a minimization problem with only $BC^*$3 unknowns, analytic gradients with respect to all $BC^*$4, gradient complexity only 4 times larger than function evaluation, and only $BC^*$5 additional memory; it further applies the method to the three-way decomposition problem and reports good convergence of Broyden algorithm (Ibragimov et al., 2017).

A second recurring pattern is multilinearization. In 3D MIMO and BD-RIS estimation, high-dimensional correlation or pilot data are arranged into structured three-way objects and then decomposed by Tucker or block Tucker models (Yuan, 2015, Almeida et al., 2024). A third pattern is channel separation by symmetry classes or information channels: in three-photon interferometry, amplitudes split into permanent, immanant, and determinant sectors (Guise et al., 2014), whereas in two-qubit Petz geometry the metric splits into population, coherence, and concurrence channels (Cho et al., 18 Sep 2025).

This suggests that the phrase “three-channel decomposition” functions as a cross-domain label for methods that isolate three structurally distinct contributions while controlling ambiguity, dimensionality, or identifiability.

2. Multivariate signal decomposition and constrained three-way factorization

In multivariate nonstationary signal analysis, the relevant model is

$BC^*$6

with $BC^*$7 individual components and $BC^*$8 complex gains (Stankovic et al., 2019). The method constructs the autocorrelation matrix

$BC^*$9

followed by the eigendecomposition

$B \in {\mathbb C}^{m \times r}$0

where $B \in {\mathbb C}^{m \times r}$1, and each eigenvector is a linear combination of the signal components,

$B \in {\mathbb C}^{m \times r}$2

Component extraction is then formulated as minimization of a time-frequency concentration measure over normalized linear combinations of the eigenvectors,

$B \in {\mathbb C}^{m \times r}$3

using

$B \in {\mathbb C}^{m \times r}$4

with $B \in {\mathbb C}^{m \times r}$5 commonly adopted in practice (Stankovic et al., 2019).

For the three-channel case $B \in {\mathbb C}^{m \times r}$6, the summary states that, as long as $B \in {\mathbb C}^{m \times r}$7, the method applies universally—independent of the particular forms of component overlap. The iterative extraction procedure computes the eigenvectors, searches the coefficient space $B \in {\mathbb C}^{m \times r}$8 for the minimum concentration combination, extracts the corresponding component, and then deflates the remaining eigenvectors by a Gram-Schmidt-like orthogonalization (Stankovic et al., 2019). The paper further states that robustness to additive noise increases with the number of sensors and that successful separation is correlated with large eigenvalue gaps between the smallest signal eigenvalue and the largest noise eigenvalue, while channel dissimilarity remains essential (Stankovic et al., 2019).

The constrained factorization framework of (Ibragimov et al., 2017) is complementary. There, analytic differentiation is propagated through Modified Gram-Schmidt, avoiding finite differences and the memory cost of Baur-Strassen. The paper explicitly recommends Modified Gram-Schmidt for better numerical stability and reports rapid and reliable convergence when the analytic gradient is used within quasi-Newton-like methods such as Broyden’s (Ibragimov et al., 2017). Taken together, these two lines of work instantiate three-channel decomposition either as sensor-domain separation by eigenstructure and concentration minimization or as parameter-reduced three-way factor fitting with analytic gradients.

3. Tucker-type three-channel decompositions in wireless communications

In 3D MIMO rotated codebook design under spatially correlated channels, the rotation matrix is approximated by a Tucker decomposition into three low-dimensional units: statistical channel direction information in horizontal and vertical directions, and statistical channel power in the joint horizontal and vertical direction (Yuan, 2015). The structured approximation is

$B \in {\mathbb C}^{m \times r}$9

where $C \in {\mathbb C}^{n \times r}$0 captures vertical statistical channel directions, $C \in {\mathbb C}^{n \times r}$1 horizontal statistical channel directions, and $C \in {\mathbb C}^{n \times r}$2 joint statistical channel power, with $C \in {\mathbb C}^{n \times r}$3 (Yuan, 2015). The induced codeword is

$C \in {\mathbb C}^{n \times r}$4

The summary states that this can be viewed as a “universal three-channel decomposition,” and that the structure is applicable for any antenna array configuration because it depends on reshaping and SVDs rather than antenna geometry (Yuan, 2015).

The same paper gives a closed-form suboptimal construction. The channel vector is reshaped as $C \in {\mathbb C}^{n \times r}$5, the left and right correlation matrices

$C \in {\mathbb C}^{n \times r}$6

are diagonalized to obtain $C \in {\mathbb C}^{n \times r}$7 and $C \in {\mathbb C}^{n \times r}$8, the projected channel is $C \in {\mathbb C}^{n \times r}$9, and the joint power vector is

$B$0

A Kronecker-product approximation and SVD then provide a closed-form suboptimal solution that reduces computational complexity (Yuan, 2015).

A more explicitly tensorial communication-theoretic realization appears in BD-RIS channel estimation. There, the received pilot signals are arranged as a third-order tensor $B$1 whose frontal slices satisfy

$B$2

and the cascaded estimation problem is recast as a block Tucker decomposition problem (Almeida et al., 2024). Two estimators are given. The first is a closed-form block Tucker Kronecker factorization (BTKF), based on the 3-mode unfolding,

$B$3

followed by parallel nearest Kronecker product approximations for each block, each reducible to a best rank-one matrix approximation via SVD (Almeida et al., 2024). The second is block Tucker alternating least squares (BTALS), which alternates least-squares updates for $B$4 and $B$5 using 1-mode and 2-mode unfoldings: $B$6

$B$7

The summary states that BTKF enjoys fast and parallel extraction in closed form and a noise rejection gain over least squares, whereas BTALS allows significantly reduced training overhead and more flexible training design (Almeida et al., 2024).

4. Quantum-information formulations

In the convex decomposition of dimension-altering quantum channels, the central conjecture attributed to Ruskai is that any $B$8-channel $B$9 can be written as

$\sigma_1,\dots,\sigma_k$0

where the generalized extreme channels have Kraus rank $\sigma_1,\dots,\sigma_k$1 (Wang, 2015). The paper states that the cases $\sigma_1,\dots,\sigma_k$2 and $\sigma_1,\dots,\sigma_k$3 for arbitrary $\sigma_1,\dots,\sigma_k$4 are proved, while for higher dimensions $\sigma_1,\dots,\sigma_k$5 and for dimension-altering channels numerical evidence supports the conjecture (Wang, 2015). In that sense, the three-channel case appears as the $\sigma_1,\dots,\sigma_k$6 instance of a universal $\sigma_1,\dots,\sigma_k$7-decomposition. The same work develops explicit circuit constructions—Ansatz I based on cosine-sine decomposition, Ansatz II motivated by higher-order generalized singular value decomposition, and Ansatz III based on simple multiplexer circuits with

$\sigma_1,\dots,\sigma_k$8

and reports low-dimensional numerical decompositions with trace-distance errors typically between $\sigma_1,\dots,\sigma_k$9 and $k \ll mr$0, depending on dimension and ansatz (Wang, 2015).

A distinct quantum-information usage appears in the support-projected Petz geometry of pure two-qubit families. The support-projected Petz metric is

$k \ll mr$1

where $k \ll mr$2 is the Petz quantum Fisher information tensor and $k \ll mr$3 the Riesz projector onto its active spectral subspace (Cho et al., 18 Sep 2025). The first main theorem proves a universal three-channel decomposition for every Petz monotone metric on any smooth two-parameter slice: $k \ll mr$4 Here $k \ll mr$5 is the population of the $k \ll mr$6 state in the reduced density matrix, $k \ll mr$7 the coherence term, and $k \ll mr$8 the concurrence (Cho et al., 18 Sep 2025). The three channels are therefore the population channel $k \ll mr$9, the coherence channel $B$0, and the concurrence channel $B$1. The same paper proves that neither the slice Gaussian curvature nor the ambient scalar curvature of the support-projected metric can, on any nonempty open set, be written as functions solely of concurrence or of the one-qubit entropy (Cho et al., 18 Sep 2025). In this setting, three-channel decomposition is not a factorization of data but an intrinsic decomposition of information geometry.

5. Symmetry-theoretic, algebraic, and identifiability perspectives

For three-channel passive optical interferometers, the action of the interferometer is described by a $B$2 unitary matrix $B$3 with $B$4 (Guise et al., 2014). The three single-photon product space decomposes as

$B$5

corresponding respectively to fully symmetric, mixed-symmetry, and fully antisymmetric sectors. The associated amplitudes can be written as

$B$6

so the coincidence landscape is decomposed into contributions governed by the permanent, immanant, and determinant (Guise et al., 2014). When all photons are simultaneous, only the permanent survives and the coincidence rate is proportional to $B$7 (Guise et al., 2014). The paper explicitly interprets this as a generalization of the Hong-Ou-Mandel effect from two channels to valleys and plateaus in three-channel interferometry.

The identifiability problem for three-way tensor decomposition is formalized in the uniqueness theory of 3-tensors. For

$B$8

Condition U requires that for any $B$9,

$C = A^* B (B^* B)^{-1},$0

where $C = A^* B (B^* B)^{-1},$1 is the number of nonzero entries of $C = A^* B (B^* B)^{-1},$2 and $C = A^* B (B^* B)^{-1},$3 (Gubkin, 2021). If the $C = A^* B (B^* B)^{-1},$4 are nonzero and pairwise non-parallel, and Condition U holds, then $C = A^* B (B^* B)^{-1},$5 has tensor rank exactly $C = A^* B (B^* B)^{-1},$6 and the decomposition is unique up to trivial rescaling and permutation (Gubkin, 2021). The paper presents this as an answer to a conjecture of Lovitz and Petrov and describes it as addressing the universal three-channel decomposition problem for 3-tensors.

A further algebraic perspective appears in the classification of unital decompositions of $C = A^* B (B^* B)^{-1},$7 into a direct vector-space sum of two subalgebras, one containing the identity: $C = A^* B (B^* B)^{-1},$8 The summary states that, up to automorphism and transpose, there are 71 inequivalent unital decompositions, excluding the trivial case $C = A^* B (B^* B)^{-1},$9 (Gubarev, 2020). Although this is a two-summand decomposition rather than a literal three-channel model, the provided summary explicitly relates it to universal three-channel decompositions through channel-theoretic and Rota-Baxter interpretations (Gubarev, 2020).

6. Three-part image decomposition and broader conceptual implications

The image-processing literature supplies a neighboring but distinct construction: directional global three-part decomposition (DG3PD) (Thai et al., 2015). The model decomposes an image $$\min_{\sigma_1,\dots,\sigma_k} \|A - B(B^*B)^{-1}B^*A\|_F,$$0 into cartoon, texture, and residual components,

$$\min_{\sigma_1,\dots,\sigma_k} \|A - B(B^*B)^{-1}B^*A\|_F,$$1

by solving

$$\min_{\sigma_1,\dots,\sigma_k} \|A - B(B^*B)^{-1}B^*A\|_F,$$2

subject to a curvelet-domain residual constraint and the reconstruction identity (Thai et al., 2015). The cartoon term is controlled by a discrete multi-directional total variation norm,

$$\min_{\sigma_1,\dots,\sigma_k} \|A - B(B^*B)^{-1}B^*A\|_F,$$3

while the texture term is controlled by a discrete multi-directional $$\min_{\sigma_1,\dots,\sigma_k} \|A - B(B^*B)^{-1}B^*A\|_F,$$4-norm defined through directional derivative synthesis over $$\min_{\sigma_1,\dots,\sigma_k} \|A - B(B^*B)^{-1}B^*A\|_F,$$5 directions (Thai et al., 2015). The optimization is solved by the augmented Lagrangian method and ADMM, with shrinkage-type subproblems and Fourier-domain updates for selected variables (Thai et al., 2015).

DG3PD is not presented as a universal three-channel decomposition in the narrow sense used in signal separation, quantum geometry, or Tucker models. A plausible implication is nevertheless that the broader decomposition literature repeatedly converges on a three-component architecture when one seeks to separate piecewise smooth structure, oscillatory structure, and residual structure, or more generally to isolate three mathematically distinct mechanisms. In the cited papers, these mechanisms are population/coherence/concurrence (Cho et al., 18 Sep 2025), permanent/immanant/determinant (Guise et al., 2014), horizontal direction/vertical direction/joint power (Yuan, 2015), or cartoon/texture/residual (Thai et al., 2015).

Across the literature, universality is therefore domain-specific. In one case it means that three sensors suffice to separate up to three overlapping components irrespective of the particular form of overlap (Stankovic et al., 2019). In another, it means applicability to arbitrary Petz monotone metrics on any smooth two-parameter slice (Cho et al., 18 Sep 2025). Elsewhere it means applicability to arbitrary antenna arrays (Yuan, 2015), or a conjectured convex decomposition for arbitrary channels in $$\min_{\sigma_1,\dots,\sigma_k} \|A - B(B^*B)^{-1}B^*A\|_F,$$6 using $$\min_{\sigma_1,\dots,\sigma_k} \|A - B(B^*B)^{-1}B^*A\|_F,$$7 generalized extreme channels (Wang, 2015). What remains stable across these otherwise heterogeneous settings is the methodological role of a three-way split: it compresses a complex object into three structurally interpretable channels while shifting the main burden of analysis to analytic gradients, eigenspaces, tensor unfoldings, symmetry sectors, or identifiability conditions.