Greedy Cofactor-Based Selection for MIMO ISAC

• The paper demonstrates how converting one-step MI loss to inverse diagonal (cofactor) ratios enables efficient RF-chain elimination in MIMO ISAC systems.
• It employs an exact MI decomposition leveraging principal-minor identities to assess redundancy and interdependence among RF chains.
• The approach outperforms conventional methods in weighted-sum MI and energy efficiency, achieving near-optimal Pareto boundaries in performance experiments.

Greedy Cofactor-Based Selection (GCS) is a backward greedy elimination method for RF-chain selection in multiple-input multiple-output integrated sensing and communication (MIMO ISAC) systems. In its explicit form, introduced in "Efficient RF Chain Selection for MIMO Integrated Sensing and Communications: A Greedy Approach" (Shin et al., 14 Jul 2025), GCS converts both communication and sensing design objectives into a unified mutual-information (MI)-based criterion, then evaluates the effect of deleting each candidate RF chain through diagonal entries of inverse matrices whose values equal principal-minor ratios and hence are directly tied to cofactors. The method is therefore a determinant-sensitive greedy procedure: it does not add chains one by one, but starts from the full set and iteratively removes the chain with the smallest weighted MI contribution.

1. Origin and problem formulation

The explicit term Greedy Cofactor-Based Selection arises in the context of transmit RF-chain selection for a bistatic MIMO ISAC system with a base station (BS) having $N_t$ antennas, a communication user equipment with $N_c$ antennas, and a sensing receiver with $N_s$ antennas (Shin et al., 14 Jul 2025). In the fully digital uniform linear array model considered there, each antenna is connected to a dedicated RF chain, so RF-chain selection is equivalent to antenna selection.

The communication channel is modeled as

$$\mathbf{H}_c = \sum_{\ell =1}^L a_{c, \ell} \mathbf{r}(\phi_{\ell}^c) \mathbf{t}^{\sf H}(\theta_{\ell}^c) \in \mathbb{C}^{N_c \times N_t},$$

and the sensing target response matrix is

$$\mathbf{H}_s = \sum_{n =1}^{N_s} a_{s, n} \mathbf{\bar r}_n \mathbf{t}^{\sf H}(\theta_n^s) \in \mathbb{C}^{N_s \times N_t}.$$

If only $K$ transmit RF chains are active, their index set is

$$\mathcal{N}_t = \{n_1,\ldots,n_K\}, \qquad |\mathcal{N}_t|=K,$$

with selection matrix

$$\mathbf{S}(\mathcal{N}_t) = [\mathbf{e}_{n_1},\mathbf{e}_{n_2},\ldots,\mathbf{e}_{n_K}] \in \mathbb{B}^{N_t\times K}.$$

The selected communication and sensing channels are then

$$\mathbf{H}_c(\mathcal{N}_t)=\mathbf{H}_c\mathbf{S}(\mathcal{N}_t),\qquad \mathbf{H}_s(\mathcal{N}_t)=\mathbf{H}_s\mathbf{S}(\mathcal{N}_t).$$

The central modeling move is to evaluate both communication and sensing by MI. With Gaussian signaling

$$\mathbf{x}(t)\sim \mathcal{CN}(\mathbf{0},P\mathbf{I}_{N_t}),$$

equal communication and sensing noise variance $N_c$0, and average transmit SNR

$N_c$1

the weighted RF-chain selection problem is formulated as

$N_c$2

with

$N_c$3

The paper characterizes this as an NP-hard subset selection problem because exhaustive search would require evaluation of all $N_c$4 possibilities (Shin et al., 14 Jul 2025).

2. Mutual-information decomposition and the cofactor identity

The communication MI for a selected subset is

$N_c$5

and, by the Weinstein-Aronszajn identity,

$N_c$6

For sensing, under the approximation

$N_c$7

the MI becomes

$N_c$8

GCS is built by expressing the effect of deleting one RF chain as a determinant ratio. For a current remaining set $N_c$9, define

$N_s$0

and

$N_s$1

Then

$N_s$2

If the $N_s$3-th RF chain is removed, the corresponding row and column are removed from $N_s$4 and $N_s$5, so the determinant after deletion is a principal minor of the original matrix. The core identity is

$N_s$6

and similarly

$N_s$7

Thus diagonal inverse entries are exactly principal-minor ratios, and therefore cofactor quantities in normalized form.

This yields the exact one-step MI decompositions

$N_s$8

$N_s$9

where

$$\mathbf{H}_c = \sum_{\ell =1}^L a_{c, \ell} \mathbf{r}(\phi_{\ell}^c) \mathbf{t}^{\sf H}(\theta_{\ell}^c) \in \mathbb{C}^{N_c \times N_t},$$0

This is the mathematical core of GCS: the one-step loss from deleting chain $$\mathbf{H}_c = \sum_{\ell =1}^L a_{c, \ell} \mathbf{r}(\phi_{\ell}^c) \mathbf{t}^{\sf H}(\theta_{\ell}^c) \in \mathbb{C}^{N_c \times N_t},$$1 is encoded exactly by inverse diagonals, not approximated heuristically (Shin et al., 14 Jul 2025).

3. Backward greedy elimination algorithm

GCS is a deletion algorithm rather than a constructive forward-selection method. It initializes with all $$\mathbf{H}_c = \sum_{\ell =1}^L a_{c, \ell} \mathbf{r}(\phi_{\ell}^c) \mathbf{t}^{\sf H}(\theta_{\ell}^c) \in \mathbb{C}^{N_c \times N_t},$$2 RF chains active,

$$\mathbf{H}_c = \sum_{\ell =1}^L a_{c, \ell} \mathbf{r}(\phi_{\ell}^c) \mathbf{t}^{\sf H}(\theta_{\ell}^c) \in \mathbb{C}^{N_c \times N_t},$$3

computes the inverse matrices

$$\mathbf{H}_c = \sum_{\ell =1}^L a_{c, \ell} \mathbf{r}(\phi_{\ell}^c) \mathbf{t}^{\sf H}(\theta_{\ell}^c) \in \mathbb{C}^{N_c \times N_t},$$4

and extracts the diagonal parameters

$$\mathbf{H}_c = \sum_{\ell =1}^L a_{c, \ell} \mathbf{r}(\phi_{\ell}^c) \mathbf{t}^{\sf H}(\theta_{\ell}^c) \in \mathbb{C}^{N_c \times N_t},$$5

At iteration $$\mathbf{H}_c = \sum_{\ell =1}^L a_{c, \ell} \mathbf{r}(\phi_{\ell}^c) \mathbf{t}^{\sf H}(\theta_{\ell}^c) \in \mathbb{C}^{N_c \times N_t},$$6, each remaining RF chain $$\mathbf{H}_c = \sum_{\ell =1}^L a_{c, \ell} \mathbf{r}(\phi_{\ell}^c) \mathbf{t}^{\sf H}(\theta_{\ell}^c) \in \mathbb{C}^{N_c \times N_t},$$7 is scored by

$$\mathbf{H}_c = \sum_{\ell =1}^L a_{c, \ell} \mathbf{r}(\phi_{\ell}^c) \mathbf{t}^{\sf H}(\theta_{\ell}^c) \in \mathbb{C}^{N_c \times N_t},$$8

The selected deletion is

$$\mathbf{H}_c = \sum_{\ell =1}^L a_{c, \ell} \mathbf{r}(\phi_{\ell}^c) \mathbf{t}^{\sf H}(\theta_{\ell}^c) \in \mathbb{C}^{N_c \times N_t},$$9

and the remaining set is updated by

$$\mathbf{H}_s = \sum_{n =1}^{N_s} a_{s, n} \mathbf{\bar r}_n \mathbf{t}^{\sf H}(\theta_n^s) \in \mathbb{C}^{N_s \times N_t}.$$0

The iteration stops when

$$\mathbf{H}_s = \sum_{n =1}^{N_s} a_{s, n} \mathbf{\bar r}_n \mathbf{t}^{\sf H}(\theta_n^s) \in \mathbb{C}^{N_s \times N_t}.$$1

The efficiency of GCS comes from updating inverse matrices by a Schur-complement formula rather than recomputing them from scratch. After permuting the removed index to the last row and column,

$$\mathbf{H}_s = \sum_{n =1}^{N_s} a_{s, n} \mathbf{\bar r}_n \mathbf{t}^{\sf H}(\theta_n^s) \in \mathbb{C}^{N_s \times N_t}.$$2

the reduced inverse is

$$\mathbf{H}_s = \sum_{n =1}^{N_s} a_{s, n} \mathbf{\bar r}_n \mathbf{t}^{\sf H}(\theta_n^s) \in \mathbb{C}^{N_s \times N_t}.$$3

with an identical update for each $$\mathbf{H}_s = \sum_{n =1}^{N_s} a_{s, n} \mathbf{\bar r}_n \mathbf{t}^{\sf H}(\theta_n^s) \in \mathbb{C}^{N_s \times N_t}.$$4. The new diagonal entries are then refreshed and the next deletion is chosen (Shin et al., 14 Jul 2025).

Because the per-step decomposition is exact, the approximation in GCS lies in the greedy strategy itself, not in the computation of one-step MI loss. The method removes the RF chain with the lowest current weighted contribution, but does not solve the full cardinality-constrained problem globally.

4. Mathematical interpretation and relation to neighboring greedy methods

Within its original setting, GCS is an explicitly cofactor-based greedy rule. Its ranking statistics are inverse diagonals, those inverse diagonals are principal-minor ratios, and the objective is a weighted sum of log-determinants. The paper therefore frames GCS as a method that accounts for interdependence among RF chains rather than only their individual strength. In the Pareto-boundary discussion, the determinant of the matrix obtained by removing a row and column is interpreted as reflecting the degree of orthogonality among the remaining channel or target-response vectors, so sequentially removing the RF chain associated with the largest such determinant is equivalent to eliminating the most redundant RF chain (Shin et al., 14 Jul 2025).

This explicit cofactor formulation distinguishes GCS from several nearby greedy paradigms. "Data-driven Vector-measurement-sensor Selection based on Greedy Algorithm" (Saito et al., 2019) is determinant- and volume-oriented, but its rule is expressed through Gram-Schmidt orthogonalization and hypervolume maximization of vector-sensor row blocks, not through cofactors. "Globally-Optimal Greedy Experiment Selection for Active Sequential Estimation" (Li et al., 2024) develops GI0 and GI1 as greedy inverse-Fisher-information rules, and the $$\mathbf{H}_s = \sum_{n =1}^{N_s} a_{s, n} \mathbf{\bar r}_n \mathbf{t}^{\sf H}(\theta_n^s) \in \mathbb{C}^{N_s \times N_t}.$$5 case admits a determinant or log-determinant interpretation, but the paper does not define a method called GCS. "Sparse inverse Cholesky factorization of dense kernel matrices by greedy conditional selection" (Huan et al., 2023) is especially close in spirit because its scalar score is a one-step Schur-complement decrement in posterior variance and its multi-target score is a log-determinant ratio, but it is specialized to symmetric positive-definite kernel matrices and is described in terms of conditional mutual information, posterior variance, and partial Cholesky factors rather than RF-chain deletion.

A plausible implication is that GCS belongs to a broader family of greedy procedures driven by determinant increments, Schur complements, or conditional variance reductions. What remains specific to the explicit GCS formulation is the backward elimination workflow and the use of principal-minor identities on the communication and sensing MI matrices of MIMO ISAC (Shin et al., 14 Jul 2025).

5. Extensions, empirical behavior, and beamspace interpretation

The original paper proposes two greedy methods, greedy eigen-based selection (GES) and GCS, and then extends the same logic to beam selection in beamspace MIMO ISAC. With a fixed analog beamforming matrix $$\mathbf{H}_s = \sum_{n =1}^{N_s} a_{s, n} \mathbf{\bar r}_n \mathbf{t}^{\sf H}(\theta_n^s) \in \mathbb{C}^{N_s \times N_t}.$$6, the effective channel and sensing covariance become

$$\mathbf{H}_s = \sum_{n =1}^{N_s} a_{s, n} \mathbf{\bar r}_n \mathbf{t}^{\sf H}(\theta_n^s) \in \mathbb{C}^{N_s \times N_t}.$$7

and the MI formulas retain the same algebraic form, so GCS applies directly in beamspace (Shin et al., 14 Jul 2025).

The paper also introduces diagonal beam selection (DBS) as an asymptotic simplification. Under the asymptotic conditions $$\mathbf{H}_s = \sum_{n =1}^{N_s} a_{s, n} \mathbf{\bar r}_n \mathbf{t}^{\sf H}(\theta_n^s) \in \mathbb{C}^{N_s \times N_t}.$$8, $$\mathbf{H}_s = \sum_{n =1}^{N_s} a_{s, n} \mathbf{\bar r}_n \mathbf{t}^{\sf H}(\theta_n^s) \in \mathbb{C}^{N_s \times N_t}.$$9, $K$0, and distinct angles of departure and arrival, the effective beamspace matrices become approximately diagonal. In that regime,

$K$1

so contributions do not change during selection and all $K$2 beams can be selected in one shot. The paper therefore presents DBS as a diagonalized or asymptotic version of GCS.

Empirically, the reported metrics are the weighted sum of normalized MI, energy efficiency based on normalized MI divided by circuit power, and the Pareto boundary between normalized communication MI and normalized sensing MI. In the weighted-sum MI experiments, GES and GCS exhibit performance nearly identical to exhaustive search. At $K$3 dB average SNR, GES/GCS outperform Random by $K$4 and Fixed by $K$5. In the energy-efficiency experiments, GCS, GES, and DBS outperform Random and improve over Full-Selection by up to $K$6. In the Pareto-boundary plots, GES and GCS achieve the best Pareto boundaries, which the paper attributes to their ability to account for determinant-based interdependence among RF chains rather than selecting high-power but redundant chains (Shin et al., 14 Jul 2025).

6. Scope, assumptions, and common points of confusion

Several assumptions delimit the scope of GCS. First, the unified objective relies on using MI for both communication and sensing. The paper presents this as a principled way to avoid the mismatch between communication metrics such as rate or MI and sensing metrics such as beam-pattern MSE, CRLB, or FIM-derived criteria. Second, the sensing MI derivation uses the approximation

$K$7

so the formulation assumes $K$8. Third, while the per-step deletion loss is exact, the overall selection remains greedy rather than globally optimal.

A common misconception is to treat GCS as a generic label for any determinant-greedy subset-selection rule. The literature summarized here does not support that identification. Projection-based greedy column subset selection (Altschuler et al., 2016), regularized greedy column subset selection (Ordozgoiti et al., 2018), and block volume-maximizing vector sensor selection (Saito et al., 2019) are all closely related theoretical neighbors, but they do not formulate their selection rules through cofactors or principal-minor ratios. Conversely, some methods outside MIMO ISAC can be interpreted cofactor-wise—such as greedy conditional selection in sparse inverse Cholesky factorization or determinant-oriented special cases of greedy experiment selection—but those papers do not explicitly define GCS under that name (Huan et al., 2023).

Another practical caveat is numerical. In theory, the matrices $K$9 and $$\mathcal{N}_t = \{n_1,\ldots,n_K\}, \qquad |\mathcal{N}_t|=K,$$0 are positive definite because they are identity plus positive semidefinite terms, so invertibility is not problematic. In implementation, however, repeated Schur-complement updates may accumulate numerical error if the matrices become ill-conditioned. The paper also notes that index bookkeeping matters because deletions are performed through permutation matrices and local matrix indices must remain consistent with the original RF-chain labels (Shin et al., 14 Jul 2025).

In that precise sense, GCS is best understood not as a generic synonym for determinant-greedy design, but as a specific MIMO ISAC backward elimination algorithm whose defining feature is the exact conversion of one-step MI loss into diagonal entries of inverse matrices, and therefore into cofactor and principal-minor information.