Two-Block Splitting Alternating Algorithm (TSAA)

• The paper introduces TSAA, a method that decomposes large-scale linear systems into two orthogonal blocks to efficiently recover sparse solutions using hard thresholding and reduced least-squares correction.
• TSAA alternates blockwise updates with explicit projections and support refinements, ensuring global convergence under a coherence-type condition using only matrix-vector products.
• Empirical results indicate that TSAA achieves rapid convergence and robust performance, outperforming traditional methods in tasks like MRI image reconstruction with fewer iterations.

Searching arXiv for recent and foundational papers on TSAA and closely related two-block splitting methods. The Two-Block Splitting Alternating Algorithm (TSAA) is a splitting alternating method for finding the sparse solution of linear systems with concatenated orthogonal matrices. In the formulation introduced in “Splitting Alternating Algorithms for Sparse Solutions of Linear Systems with Concatenated Orthogonal Matrices,” the algorithm is designed for matrices of the form $A=[\Phi_a,\Phi_b]$, decomposes a large-scale linear system into two coupled subsystems, and then combines the subsystem solutions to produce the sparse solution of the original system. Its iteration uses only matrix-vector products and reduced orthogonal projections, and it is globally convergent to the sparse solution if the matrix, together with the sparsity level of the solution, satisfies a coherence-type condition (Zhao et al., 29 Sep 2025).

1. Problem class and structural setting

TSAA is posed for the sparse recovery problem

$\min \{\|x\|_0 : y = Ax\},$

with

$$A = [\Phi_a, \Phi_b], \qquad \Phi_a,\Phi_b \in \mathbb{R}^{m\times m},$$

where the two blocks are orthogonal, $y \in \mathbb{R}^m$, and the unknown vector is partitioned as $x=(x_a,x_b)\in\mathbb{R}^{2m}$. The target is a $K$-sparse solution (Zhao et al., 29 Sep 2025).

The core structural observation is that the measurement vector can be written as

$y = y_a^* + y_b^*,$

with

$y_a^* = \Phi_a x_a^*, \qquad y_b^* = \Phi_b x_b^*.$

This recasts the original recovery task as a decomposition problem: one seeks a splitting of $y$ into two components, each associated with one orthogonal subsystem. Because $\Phi_a$ and $\min \{\|x\|_0 : y = Ax\},$0 are orthogonal, each subsystem can be inverted by an orthogonal projection, which is the principal reason the algorithm can operate through reduced projections rather than full-scale subproblem solvers (Zhao et al., 29 Sep 2025).

This structure distinguishes TSAA from generic sparse recovery procedures that treat $\min \{\|x\|_0 : y = Ax\},$1 as an unstructured sensing matrix. A plausible implication is that TSAA is most natural when the sensing operator already arrives in a concatenated orthogonal form, rather than as an arbitrary dictionary.

2. Alternating split and blockwise thresholding

The iteration alternates between the two blocks by estimating one component while keeping the other fixed. The paper uses the hard-thresholding operator $\min \{\|x\|_0 : y = Ax\},$2, which retains the $\min \{\|x\|_0 : y = Ax\},$3 largest entries in magnitude and zeros the rest, and the mutual coherence

$\min \{\|x\|_0 : y = Ax\},$4

with $\min \{\|x\|_0 : y = Ax\},$5 denoting the columns of $\min \{\|x\|_0 : y = Ax\},$6 (Zhao et al., 29 Sep 2025).

Starting from an initial vector $\min \{\|x\|_0 : y = Ax\},$7, the first block update computes

$\min \{\|x\|_0 : y = Ax\},$8

solves $\min \{\|x\|_0 : y = Ax\},$9 by orthogonality, and then applies hard thresholding: $$A = [\Phi_a, \Phi_b], \qquad \Phi_a,\Phi_b \in \mathbb{R}^{m\times m},$$0

The second block update is symmetric. Using $$A = [\Phi_a, \Phi_b], \qquad \Phi_a,\Phi_b \in \mathbb{R}^{m\times m},$$1, it forms

$$A = [\Phi_a, \Phi_b], \qquad \Phi_a,\Phi_b \in \mathbb{R}^{m\times m},$$2

solves $$A = [\Phi_a, \Phi_b], \qquad \Phi_a,\Phi_b \in \mathbb{R}^{m\times m},$$3, and thresholds: $$A = [\Phi_a, \Phi_b], \qquad \Phi_a,\Phi_b \in \mathbb{R}^{m\times m},$$4

The two-block character of TSAA is therefore literal: each alternation consists of an orthogonal projection onto one block followed by sparsity enforcement, then the same procedure for the other block. Because each block is handled through its own orthogonal factor, the updates are explicit and avoid large subproblems over the full $$A = [\Phi_a, \Phi_b], \qquad \Phi_a,\Phi_b \in \mathbb{R}^{m\times m},$$5 matrix (Zhao et al., 29 Sep 2025).

3. Recombination, active sets, and least-squares correction

TSAA does not terminate its iteration at the two thresholded block estimates. Instead, it performs a recombination and fine-tuning stage that merges blockwise information with a global residual correction. This stage is central to the algorithmic identity of TSAA (Zhao et al., 29 Sep 2025).

After computing

$$A = [\Phi_a, \Phi_b], \qquad \Phi_a,\Phi_b \in \mathbb{R}^{m\times m},$$6

the method forms the gradient-like vector

$$A = [\Phi_a, \Phi_b], \qquad \Phi_a,\Phi_b \in \mathbb{R}^{m\times m},$$7

It then builds an index set

$$A = [\Phi_a, \Phi_b], \qquad \Phi_a,\Phi_b \in \mathbb{R}^{m\times m},$$8

where $$A = [\Phi_a, \Phi_b], \qquad \Phi_a,\Phi_b \in \mathbb{R}^{m\times m},$$9, and the paper uses $y \in \mathbb{R}^m$0 to capture important entries (Zhao et al., 29 Sep 2025).

Over this support, the algorithm solves a small least-squares problem: $y \in \mathbb{R}^m$1 From $y \in \mathbb{R}^m$2, it identifies the $y \in \mathbb{R}^m$3 most significant entries,

$y \in \mathbb{R}^m$4

and computes the final correction

$y \in \mathbb{R}^m$5

This support-refinement step means that TSAA is not merely a pair of alternating hard-thresholding passes. It alternates between subsystem recovery and global consistency enforcement. The detailed summary in the source emphasizes that all heavy-duty operations are on small support sets of size at most $y \in \mathbb{R}^m$6, and that no large least-squares or expensive subroutines over the full matrix $y \in \mathbb{R}^m$7 are required per iteration (Zhao et al., 29 Sep 2025).

4. Convergence theory and coherence requirements

The convergence theory of TSAA is tied to a coherence-type condition on the matrix $y \in \mathbb{R}^m$8 and the sparsity level $y \in \mathbb{R}^m$9. The abstract states that the proposed algorithms are globally convergent to the sparse solution of a linear system if the matrix, along with the sparsity level of the solution, satisfies a coherence-type condition (Zhao et al., 29 Sep 2025).

In the detailed summary, this guarantee is sharpened through a mutual-coherence bound involving $x=(x_a,x_b)\in\mathbb{R}^{2m}$0. Under that condition, Theorem 3.6 gives a geometric decay estimate of the form

$x=(x_a,x_b)\in\mathbb{R}^{2m}$1

which establishes linear convergence to the true sparse solution (Zhao et al., 29 Sep 2025).

Two points are technically significant. First, the guarantee is global rather than local: the summary explicitly states that convergence is independent of initialization under the stated coherence regime. Second, the proof leverages the orthogonal block structure and the sparsity control introduced by hard thresholding and restricted least-squares correction. This suggests that the convergence mechanism is inseparable from the concatenated orthogonal model; it is not presented as a black-box guarantee for arbitrary dictionaries.

A common misconception is to read “two-block splitting” as a purely implementation-level decomposition. In TSAA, the two-block structure is also part of the proof architecture: the orthogonal subsystems, the coherence bound, and the support-selection steps are all coupled to the convergence statement.

5. Position within the broader two-block splitting literature

The TSAA of sparse recovery belongs to a broader research family in which a problem is reformulated into two interacting blocks and solved by alternating updates. Related work spans several distinct algorithmic traditions.

In convex optimization with linear constraints, “The Proximal Alternating Minimization Algorithm for two-block separable convex optimization problems with linear constraints” develops Proximal AMA by adding proximal terms induced by variable metrics so that the two subproblems can, for suitable choices, be reduced to proximal operators (Bitterlich et al., 2018). In convex minimization with linear equality constraints, an inertial ADMM is obtained by applying the inertial Douglas-Rachford splitting algorithm to the dual of a two-block separable problem (Yang et al., 2020). In convex bi-clustering and co-clustering, a standard two-block ADMM splits the problem into a primal block and auxiliary difference variables, while a Generalized ADMM is introduced because it avoids an expensive tensor Sylvester equation in the primal update (Weylandt, 2019). In nonconvex constrained optimization, monotone splitting SQP algorithms decompose a two-block problem with linear equality, inequality, and box constraints by combining ADMM ideas, sequential quadratic programming, slack variables, and Armijo line search (Jian et al., 2023). A separate line of work on multi-block separable convex minimization first transforms a many-block problem into an equivalent two-block problem and then solves it using standard two-block ADMM, with $x=(x_a,x_b)\in\mathbb{R}^{2m}$2 iteration complexity guarantees (Wang et al., 2013).

This body of work suggests that “two-block splitting alternating algorithm” is best understood as a structural designation rather than a single canonical iteration. In one setting it denotes hard-thresholding plus reduced least-squares over concatenated orthogonal dictionaries (Zhao et al., 29 Sep 2025); in others it denotes ADMM-, AMA-, or SQP-based decompositions over constrained convex or nonconvex programs (Bitterlich et al., 2018, Weylandt, 2019, Yang et al., 2020, Jian et al., 2023).

The broader literature also shows that two-block reformulation does not automatically imply the best practical speed. For example, the multi-block ADMM study reports that multiple-block ADMM, although lacking general convergence guarantees, typically outperforms two-block ADMMs numerically (Wang et al., 2013). Likewise, in convex co-clustering, the theoretical and experimental analysis indicates that Generalized ADMM is far more efficient for large problems than the standard two-block ADMM (Weylandt, 2019). This contextualizes TSAA’s contribution: its relevance depends on the compatibility between the two-block split and the native structure of the target problem.

6. Empirical behavior and application scope

The reported numerical results for TSAA are centered on sparse vector recovery and real MRI image reconstruction. According to the abstract, the proposed algorithms can quickly and accurately locate the sparse solution of a linear system with significantly fewer iterations than several mainstream iterative methods (Zhao et al., 29 Sep 2025).

The detailed summary states that TSAA often surpasses HTP, CoSaMP, OMP, IHT, FISTA, and $x=(x_a,x_b)\in\mathbb{R}^{2m}$3-minimization, especially with higher sparsity ratios, and that it typically finds good solutions in just a few iterations, sometimes $x=(x_a,x_b)\in\mathbb{R}^{2m}$4–$x=(x_a,x_b)\in\mathbb{R}^{2m}$5 (Zhao et al., 29 Sep 2025). It also reports that the method uses only fast matrix-vector multiplications and small least-squares fits on support sets of size at most $x=(x_a,x_b)\in\mathbb{R}^{2m}$6, and that it is robust in noisy settings. For MRI image reconstruction, the summary states that TSAA yields high PSNR and much faster run time for a given recovery accuracy (Zhao et al., 29 Sep 2025).

These observations should be read together with the structural assumptions. The favorable behavior is reported for linear systems with concatenated orthogonal matrices, precisely the setting for which the algorithm was designed. A plausible implication is that the empirical strengths of TSAA—few iterations, low per-iteration complexity, and accurate support localization—derive from that alignment between block structure and algorithmic split, rather than from a universally dominant sparse-recovery strategy.