Row–Column Hybrid Grouping

• Row–Column Hybrid Grouping is a design pattern that integrates both row and column structures to optimize segmentation, redundancy distribution, and decomposition across arrays.
• It underpins diverse applications from experimental design and ultrasound imaging to matrix recovery and attention models by leveraging two-dimensional interactions.
• The methodology overcomes limitations of one-dimensional approaches by coupling both axes, thereby enhancing fault tolerance, spatial resolution, and computational efficiency.

Row–Column Hybrid Grouping denotes a family of methods that exploit structure, redundancy, or decomposition simultaneously across the row and column dimensions of an array, matrix, grid, or table. In the literature, the term and closely related constructions arise in several distinct senses: simultaneous segmentation of rows and columns in binary matrices with co-variables; row–column blocking and augmentation in experimental design; orthogonal sub-aperture activation in bias-switchable ultrasound arrays; redundancy distributed across both rows and columns for analog in-memory computing; and alternating row/column partitioning for blocked sparse-matrix storage. The common feature is that performance is obtained by coupling the two axes rather than optimizing only a row-wise or column-wise representation (Iovleff et al., 2018, Piepho et al., 25 Nov 2025, Jeon et al., 21 Aug 2025, Roozkhosh et al., 2021).

1. Conceptual scope and recurring pattern

Across the cited literature, row–column hybrid grouping is not a single standardized algorithm. Rather, it is a recurring design pattern in which the row axis and the column axis are both treated as active degrees of freedom. In some settings this means simultaneous latent clustering of rows and columns; in others it means distributing redundancy over a two-dimensional hardware substrate; in others it means decomposing a dense operation into row- and column-specific components that interact only at crossings. This suggests that the phrase is best understood as a cross-domain methodological motif rather than a domain-specific formalism.

Domain	Grouped object	Representative mechanism
Experimental design	Rectangular treatment layouts	Replicated checks across rows and columns; contractions
Statistical learning	Data matrices	Joint row/column latent clustering with side information
Ultrasound imaging	Orthogonal apertures	Bias-switched row/column activation; Hadamard encoding
Analog IMC	Weight bitmaps on arrays	Redundancy across both rows and columns
Sparse linear algebra	Sparse matrices	Alternating contiguous row/column partitioning
Dense prediction	Feature maps	Decomposed row and column queries
Character tables	Rows and columns of irreducible data	Dual TQFT constructions and partial-sum groupings

A second recurring feature is that hybridization is usually introduced to address a concrete failure mode of one-dimensional organization. In the design literature, row-only or column-only blocking cannot control two orthogonal nuisance sources. In RCA imaging, a single long-aperture direction leaves line-shaped PSFs and shadow constraints. In IMC arrays, conventional column grouping is vulnerable to stuck-at faults in high-significance bits. In VBR storage, heuristics that merge rows and columns jointly lack optimality guarantees because full two-dimensional grouping is NP-hard (Piepho et al., 25 Nov 2025, Palamar et al., 12 Jun 2025, Jeon et al., 21 Aug 2025, Ahrens et al., 2020).

2. Experimental-design formulations

In row–column experimental design, a design arranges $n=v\times s$ experimental units in $v$ rows and $s$ columns, with rows and columns serving as two orthogonal blocking factors. The usual linear model is

$y = X\tau + Z_R\rho + Z_C\kappa + \epsilon,$

where $\tau$ are fixed treatment effects, $\rho$ row effects, $\kappa$ column effects, and $\epsilon \sim N(0,\sigma^2 I)$. Augmented row–column designs are motivated by settings, such as early-generation plant breeding, in which a small set of replicated checks is distributed across the grid while all remaining test treatments appear exactly once. Estimation of row and column effects, and an unbiased error-variance estimate, then comes entirely from the replicated checks; connectivity through the checks is critical (Piepho et al., 25 Nov 2025).

The contraction approach generates an augmented design from an auxiliary row–column design on $v$ pseudo-treatments in a $k\times s$ array, where $v$0 is the number of checks per column in the final design. The contraction is represented by a binary incidence matrix $v$1 with $v$2 if pseudo-treatment $v$3 appears in column $v$4. After labeling test lines and checks, each $v$5 in $v$6 determines the placement of a check in the augmented layout, and all remaining cells are filled by the test lines in an arbitrary but fixed order. The information matrix for treatment effects after projecting out rows and columns is

$v$7

and the canonical efficiency factors are the nontrivial eigenvalues of

$v$8

The average efficiency factor is

$v$9

Piepho and Williams’ decomposition links the augmented design to the contraction through harmonic means $s$0 and $s$1, yielding

$s$2

In the $s$3 example with $s$4, the contraction has $s$5 and $s$6, giving $s$7, while direct augmented-design search in CycDesigN gave $s$8 after a much longer search (Piepho et al., 25 Nov 2025).

A distinct but related use of row–column structure appears in constructions of $s$9fi-optimal factorial designs. There the $y = X\tau + Z_R\rho + Z_C\kappa + \epsilon,$0 runs are placed in an $y = X\tau + Z_R\rho + Z_C\kappa + \epsilon,$1 grid with no replication, and a full-row-rank generator matrix

$y = X\tau + Z_R\rho + Z_C\kappa + \epsilon,$2

over $y = X\tau + Z_R\rho + Z_C\kappa + \epsilon,$3 defines row-shifts $y = X\tau + Z_R\rho + Z_C\kappa + \epsilon,$4, column-shifts $y = X\tau + Z_R\rho + Z_C\kappa + \epsilon,$5, and treatment assignment

$y = X\tau + Z_R\rho + Z_C\kappa + \epsilon,$6

Confounding is characterized by linear dependence relations among columns of $y = X\tau + Z_R\rho + Z_C\kappa + \epsilon,$7: a main effect or two-factor interaction is unconfounded exactly when the corresponding columns satisfy the required independence conditions. The paper gives theoretical constructions of $y = X\tau + Z_R\rho + Z_C\kappa + \epsilon,$8 full factorial $y = X\tau + Z_R\rho + Z_C\kappa + \epsilon,$9fi-optimal row-column designs for any odd prime level $\tau$0 and any parameter combination, and of $\tau$1 fractional factorial $\tau$2fi-optimal row-column designs for any prime level $\tau$3 and any parameter combination; in the listed fractional cases, the resulting designs have $\tau$4fi-efficiency $\tau$5 (Zhang et al., 2023).

3. Joint clustering and matrix-structured inference

In statistical data analysis, row–column hybrid grouping often refers to simultaneous clustering of the rows and columns of a matrix into latent blocks. In the latent block model with Gaussian co-variables, the observed data are an $\tau$6 binary matrix $\tau$7 and a $\tau$8-dimensional Gaussian co-variable $\tau$9 attached to each row. Latent row-cluster indicators $\rho$0 and column-cluster indicators $\rho$1 are introduced, with $\rho$2 and $\rho$3 a priori independent. Conditional on $\rho$4,

$\rho$5

while

$\rho$6

The model is fitted with a mean-field variational EM in which $\rho$7 and the lower bound $\rho$8 is maximized by alternating row updates, column updates, Gaussian-mixture updates, and weighted logistic regressions for the block parameters $\rho$9. The per-iteration cost is $\kappa$0, plus $\kappa$1 for covariance inversions and $\kappa$2 for the logistic subproblems. In the Senegalese malaria application, with $\kappa$3 individuals and $\kappa$4 SNPs, the method identified two subject clusters and $\kappa$5 SNP blocks (Iovleff et al., 2018).

A more classical exact-recovery formulation is the jointly clustered binary-matrix model with erasures and flips. There, an $\kappa$6 matrix is block-constant with $\kappa$7 equal-sized row clusters and $\kappa$8 equal-sized column clusters, and the goal is exact recovery from a partially observed noisy version. The information-theoretic lower bound states that if

$\kappa$9

then exact recovery fails with probability at least $\epsilon \sim N(0,\sigma^2 I)$0. Three algorithmic regimes are analyzed. The combinatorial method has runtime $\epsilon \sim N(0,\sigma^2 I)$1 and, in the noiseless case, succeeds when $\epsilon \sim N(0,\sigma^2 I)$2. The convex method uses a nuclear-norm relaxation and succeeds when

$\epsilon \sim N(0,\sigma^2 I)$3

The spectral method is dominated by one SVD on an $\epsilon \sim N(0,\sigma^2 I)$4 matrix and succeeds when

$\epsilon \sim N(0,\sigma^2 I)$5

The stated comparison is a smooth time–data trade-off: exponential-time methods are statistically closest to the lower bound, while polynomial-time methods require more observations (Xu et al., 2013).

These formulations clarify an important point. In this statistical literature, “hybrid” does not merely mean that both dimensions are present in the data matrix; it means that both dimensions are assigned latent structure and that inference on one axis is coupled to inference on the other. In the co-variable model, the Gaussian $\epsilon \sim N(0,\sigma^2 I)$6 regularizes row assignments; in the exact-recovery model, row and column partitions must be recovered simultaneously because the signal is block-constant only after both clusterings are aligned.

4. Ultrasound and array-imaging realizations

In ultrasonic imaging, row–column hybrid grouping is implemented physically on bias-switchable row-column arrays. A TOBE array consists of a $\epsilon \sim N(0,\sigma^2 I)$7 element, $\epsilon \sim N(0,\sigma^2 I)$8-pitch $\epsilon \sim N(0,\sigma^2 I)$9D transducer in which each acoustic element is formed by the overlap of a row electrode on the top face and a column electrode on the bottom face. Because the electrostrictive relaxor is not piezoelectric without bias, fast switching between bias states makes four element states possible: row-only active, column-only active, both electrodes biased with the same polarity, and opposite polarities used in Hadamard coding. Each of the $v$0 rows and $v$1 columns has its own bias-tee, allowing switching in less than $v$2 (Palamar et al., 12 Jun 2025).

FORCES uses a sequence of $v$3 bias patterns and $v$4 transmit/receive events on an $v$5 array. On event $v$6, the columns follow a Hadamard row while the rows are biased to form an elevational focus at the desired depth; after $v$7 transmissions, the received data are sign-corrected and software Hadamard-decoded. The resulting synthetic summation is

$v$8

Experimentally, FORCES improved lateral FWHM from approximately $v$9 or $k\times s$0 to approximately $k\times s$1 on the $k\times s$2 array, and from approximately $k\times s$3 or $k\times s$4 to approximately $k\times s$5 on the $k\times s$6 array. For $k\times s$7 cysts at $k\times s$8 depth on the high-frequency array, FORCES achieved $k\times s$9, compared with approximately $v$00 for VLS and approximately $v$01 for TPW. The field of view extended to beyond $v$02 off-axis, whereas TPW and VLS collapsed rapidly beyond $v$03 (Palamar et al., 12 Jun 2025).

HERCULES applies the same bias-switchable architecture to expansive $v$04D scanning. If $v$05 denotes the isolated echo from element $v$06 and the $v$07th event uses the $v$08th row of an $v$09 Hadamard matrix $v$10, then the measured receive signal on column $v$11 is

$v$12

or in matrix form,

$v$13

Decoding uses

$v$14

which recovers an effective full $v$15 receive aperture over $v$16 events. The paper reports simulation and experimental implementation with comparable resolution to existing RCA imaging methods at hundreds of frames per second (Dahunsi et al., 13 Jun 2025).

A different hybrid mechanism is row-column specific Frame Multiply and Sum. In RC-FMAS, the row- and column-transmitted volumes are paired nonlinearly through the signed geometric mean

$v$17

and the final image is

$v$18

With $v$19 steering angles split as $v$20, RC-FMAS used $v$21 row–column pairs instead of $v$22 FMAS pairs. In the reported PSF experiment, lateral FWHM improved from $v$23 for DAS to $v$24 for RC-FMAS; PMSLR increased from approximately $v$25 to approximately $v$26; TNR increased from approximately $v$27 to approximately $v$28; and TCR increased from approximately $v$29 to approximately $v$30 (Hansen-Shearer et al., 2021).

5. Hardware compilation, memory systems, and sparse storage

In analog IMC, row–column hybrid grouping is a multi-bit weight representation that generalizes conventional column grouping. Conventional $v$31 grouping maps bit slices of a weight to $v$32 adjacent columns in one row. Hybrid grouping, denoted $v$33, uses $v$34 rows that share the same input voltage, so that each weight is represented by an aggregate bitmap of size $v$35. With positive and negative arrays $v$36 and $v$37, the stuck-at-fault model is

$v$38

and the decoded faulty weight is

$v$39

Theorem 1 establishes clipping: if at least one SAF exists in the $v$40 group, then the representable range is strictly smaller than in the fault-free case. Theorem 2 gives a condition for inconsecutivity gaps when all cells of a significance bit are stuck and

$v$41

The compiler reformulates fault-aware decomposition and closest value matching as ILPs, uses range and consecutivity checks to bypass most hard cases, and applies a seven-step pipeline. Reported results include up to $v$42p accuracy improvement, $v$43 faster compilation, and $v$44 energy efficiency gain compared to existing baselines; for ResNet-18 on Xeon Silver 4210, the full pipeline reduces compilation from approximately $v$45 min to approximately $v$46 s (Jeon et al., 21 Aug 2025).

Relational Memory provides a different hardware interpretation. It interposes an FPGA-based PLIM engine between CPU and DRAM and exposes an ephemeral “column group” abstraction to the CPU, while the underlying data remain stored row-wise. The FPGA decodes the alias address into a base pointer plus column offsets, issues narrow read bursts, strips unwanted bytes, repacks the desired fields into a $v$47-byte cache line, and returns that line as if the projected columns were already stored contiguously. The prototype showed access to the desired columns up to $v$48 faster than accessing them from their row-wise counterpart, parity with pure columnar access for low projectivity, and up to $v$49 speedup over column-store as projectivity increased. The same pipeline is described as extensible to hardware selection, group by, aggregation, and joins (Roozkhosh et al., 2021).

For sparse-matrix storage, VBR groups adjacent rows and columns and stores resulting nonzero blocks densely. Given contiguous row partition $v$50 and column partition $v$51, the memory model is

$v$52

The optimization of full two-dimensional grouping is NP-hard under several cost models. The paper therefore develops an optimal linear-time dynamic program for the row-only $v$53D-VBR case and an alternating heuristic that optimally repartitions one axis while holding the other fixed. On $v$54 real sparse matrices, median memory-footprint reduction versus CSR reached $v$55 for alternating $v$56D-VBR with the MinMemory objective, while the best SpMV time speedup was $v$57 for MinCompute $v$58D-VBR; alternating $v$59D-VBR with MinCompute achieved $v$60 (Ahrens et al., 2020).

These systems papers make explicit a practical distinction. Hybrid grouping is often introduced not to maximize symmetry for its own sake, but to reallocate cost. In IMC it shifts fault sensitivity from a single high-significance column into a two-dimensional bitmap. In near-memory query processing it shifts row-to-column reorganization from software into hardware. In sparse storage it trades exact two-dimensional optimality for alternating one-dimensional optimizations.

6. Decompositions, dualities, and generalized row–column formalisms

A generalized algorithmic form of row–column hybridization appears in dense prediction. DFlatFormer replaces $v$61 dense decoder queries with two disjoint learnable sets: $v$62 row queries and $v$63 column queries. The encoder feature map is flattened twice, once row-wise and once column-wise, to align keys and values with the corresponding queries. The resulting complexity drops from

$v$64

to

$v$65

Grouping and pooling reduce the attention cost further to

$v$66

where $v$67 and $v$68. Row–column interaction is then introduced through cross-attention:

$v$69

and the final full-resolution map is reconstructed by

$v$70

Here hybrid grouping is neither clustering nor redundancy; it is a structural decomposition of a dense operator into mutually interacting row and column subproblems (Wang et al., 2022).

A measurement-theoretic variant appears in low-rank matrix recovery from row-and-column affine measurements. The unknown matrix $v$71 is observed through

$v$72

with total number of measurements

$v$73

The SVLS algorithm computes the top-$v$74 singular vectors from one side, solves a least-squares problem using the opposite-side measurements, repeats symmetrically, and selects the better fit. In the noiseless Gaussian Rows & Columns setting, if $v$75, then SVLS recovers $v$76 exactly with probability one. In the noisy GRC case with $v$77, the reconstruction obeys

$v$78

with high probability (Wagner et al., 2015).

An algebraic version is developed through row–column duality in finite-group character tables. On one side, $v$79-TQFTs based on the center $v$80 yield integrality theorems for partial column sums such as

$v$81

On the dual side, $v$82-TQFTs based on the representation ring yield integrality of partial row sums such as

$v$83

The two are related by a Frobenius-algebra duality exchanging rows and columns, which the paper presents as a unified “grid” of integral partial sums indexed by subsets defined through level sets of normalized characters or characters themselves (Padellaro et al., 2023).

Taken together, these examples support a precise negative statement: row–column hybrid grouping is not synonymous with ordinary co-clustering. The literature includes latent block models, affine measurement schemes, decomposed query architectures, fault-tolerant array encodings, and algebraic dualities. What is shared is the deliberate exploitation of two-dimensional separability or two-dimensional coupling. What differs is the object being grouped: treatments, users and variables, sub-apertures, conductance cells, sparse-matrix indices, attention queries, or character-table entries.