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arxiv: 2508.15026 · v2 · submitted 2025-08-20 · 🧮 math.OC

Basis pursuit by inconsistent alternating projections

Roger Behling , Yunier Bello-Cruz , Luiz-Rafael Santos , Paulo J. S. Silva This is my paper

Pith reviewed 2026-05-18 21:40 UTC · model grok-4.3

classification 🧮 math.OC
keywords basis pursuitalternating projectionsinconsistent projectionslinear convergenceℓ1 minimizationsparse recoveryerror boundsconvex optimization
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The pith

Basis pursuit is solved by alternating projections on deliberately inconsistent subproblems that converge linearly.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a direct method for basis pursuit that minimizes the ℓ1-norm of solutions to a linear system without first rewriting the problem as a linear program. Subproblems are constructed so their feasible sets are disjoint by design, deliberately creating inconsistency in the alternating projections scheme. The authors establish that the sequence of ℓ1-radii converges linearly to the optimal norm value. When a unique solution exists, the generated points converge linearly to that solution at a rate set by an error bound between the feasible set and the optimal ℓ1-ball. The approach performs competitively with existing solvers on both synthetic and real data.

Core claim

By enforcing inconsistency in alternating projections subproblems for the ℓ1-minimization form of basis pursuit, the ℓ1-radii converge linearly to the optimal value, and when the solution is unique, all generated sequences converge linearly to it at a rate governed by a natural error bound between the feasible set and the optimal ℓ1-ball.

What carries the argument

The inconsistent alternating projections scheme, with subproblems deliberately constructed to have disjoint feasible sets.

If this is right

  • The ℓ1-radii converge linearly to the optimal value.
  • When the solution is unique, the generated sequences converge linearly to the solution.
  • The convergence rate is controlled by the natural error bound between the feasible set and the optimal ℓ1-ball.
  • The method is numerically competitive against state-of-the-art open-source solvers on synthetic and real-world instances.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The deliberate inconsistency technique might accelerate projection methods on other convex problems that are usually solved via linear programming reformulations.
  • Implementation on large-scale compressed sensing instances could show whether the linear rate translates into practical speed gains over interior-point or first-order LP solvers.
  • The error-bound analysis may extend to related sparse recovery formulations with additional convex constraints.

Load-bearing premise

Subproblems can be constructed with disjoint feasible sets by design, and a natural error bound exists between the feasible set and the optimal ℓ1-ball.

What would settle it

A numerical run on an instance with unique solution in which the observed convergence rate of the radii or iterates deviates from the linear rate predicted by the error bound.

Figures

Figures reproduced from arXiv: 2508.15026 by Luiz-Rafael Santos, Paulo J. S. Silva, Roger Behling, Yunier Bello-Cruz.

Figure 1
Figure 1. Figure 1: shows the performance profile [Dolan and Moré 2002] comparing these two variants in the Lorenz; see a detailed description below. It is clear from this experiment that using HOC greatly accelerates the convergence of BPMap and, hence, it will be adopted as the default from now on. 2 0 2 1 2 2 2 3 2 4 0.00 0.25 0.50 0.75 1.00 Within this factor of the best (log scale) Proportion of problems BPMap BPMap-HOC … view at source ↗
Figure 2
Figure 2. Figure 2: shows the performance profile of the HOC variants of BPMap and BPMap-BIN. The binary variant is faster. Hence, we will use this version in the comparison with the other methods below. 2 0.0 2 0.5 2 1.0 2 1.5 2 2.0 2 2.5 0.00 0.25 0.50 0.75 1.00 Within this factor of the best (log scale) Proportion of problems BPMap-HOC BPMap-HOC-Bin [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: shows the performance profile comparing BPMap-HOC-BIN with ISAL1 and the LP formulation solved using HiGHS. As one can see, BPMap-HOC-BIN comes out as a clear winner against ISAL1. It also surprisingly outperforms HiGHS. However, in this case, we see a fast increase in the HiGHS profile starting at 2 4 . This suggests that HiGHS is performing well in a subgroup of the problems. 2 0 2 1 2 2 2 3 2 4 2 5 2 6 … view at source ↗
Figure 4
Figure 4. Figure 4: Performance profile comparing BPMap-HOC-BIN with ISAL1 and the LP formulation solved with HiGHS the Lorenz test set. The top line show the comparison for problems where at least one of the matrix dimensions is smaller than 2048. The second line is the results when both matrix dimensions are 2048 or more. The right-hand sides are then obtained by multiplying the data matrix 𝐴 by the solution that has the sp… view at source ↗
Figure 5
Figure 5. Figure 5: Performance profile comparing BPMap-HOC-BIN with ISAL1 and the LP formulation solved with HiGHS in full the LSS test set. can also benefit from moving the solution of the linear systems to a GPU to compensate for the advantages that HiGHS showed on the sparse problems in the Lorenz test set. The computer node is equipped with 2 NVidia L40S GPUs. Only one GPU was used in the tests [PITH_FULL_IMAGE:figures/… view at source ↗
Figure 6
Figure 6. Figure 6: Performance profile comparing BPMap-HOC-BIN using GPU (NVidia L40S) with ISAL1 and the LP formulation solved with HiGHS in the problems from LSS test set with at least 1, 000 rows and 10, 000 columns. of expanding ℓ1-balls. We showed that the radii converge linearly to the optimal ℓ1-norm and that all cluster points of the generated sequence are solutions to the problem. Practical variants BPMap-HOC, incor… view at source ↗
read the original abstract

Basis pursuit is the problem of finding a vector with smallest $\ell_1$-norm among the solutions of a given linear system of equations. It is a well-known convex relaxation of the sparse affine feasibility problem, where sparse solutions to underdetermined systems are sought. Since basis pursuit admits a linear programming reformulation, standard LP solvers are directly applicable. We instead address the basis pursuit directly in its $\ell_1$-minimization form, without LP reformulation, via a scheme that uses alternating projections in its subproblems. These subproblems are designed to be inconsistent in the sense that they relate to two non-intersecting sets. Recently in [R. Behling, Y. Bello-Cruz and L.-R. Santos, SIAM J. Optim., 31 (2021), pp. 2863-2892], inconsistency coming from infeasibility has been shown to accelerate convergence of alternating projections. We deliberately enforce this inconsistency by constructing subproblems whose feasible sets are disjoint by design. We prove that the resulting $\ell_1$-radii converge linearly to the optimal value, and that when the solution is unique, all generated sequences converge linearly to it at a rate governed by a natural error bound between the feasible set and the optimal $\ell_1$-ball. The proposed method is numerically competitive against state-of-the-art open-source solvers on synthetic and real-world instances.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proposes solving basis pursuit (minimum ℓ1-norm solution to Ax=b) via alternating projections applied to a sequence of deliberately inconsistent subproblems whose feasible sets are constructed to be disjoint. It claims to prove that the generated ℓ1-radii converge linearly to the optimal value, and that when the solution is unique all iterates converge linearly to it at a rate controlled by a natural error bound between each feasible set and the optimal ℓ1-ball. Numerical tests indicate competitiveness with state-of-the-art open-source LP solvers on synthetic and real instances.

Significance. If the linear-convergence claims can be placed on a fully rigorous footing with an explicit, instance-independent positive lower bound on the governing error-bound constant, the work would supply a projection-based alternative to LP reformulations of basis pursuit and would illustrate a concrete use of inconsistency-driven acceleration for a canonical sparse-recovery problem.

major comments (2)
  1. [§4 (Convergence Analysis)] §4 (Convergence Analysis), main theorem on linear rates: the linear convergence of both radii and (unique-solution) iterates is asserted to follow from a 'natural error bound' between the constructed feasible set and the optimal ℓ1-ball, yet no explicit positive lower bound on this constant is derived in terms of A, b or the current radius. Without such a bound the claimed uniform linear rate is not guaranteed and the central convergence result remains conditional on an unverified assumption.
  2. [§3 (Subproblem construction)] §3 (Subproblem construction): the manuscript states that the subproblems are designed so their feasible sets are disjoint 'by construction,' but does not supply a self-contained verification that this design preserves equivalence to the original basis-pursuit problem and does not inadvertently make the error-bound constant vanish for feasible (A,b) pairs.
minor comments (2)
  1. [§2] The phrase 'natural error bound' is used repeatedly before it is formally defined; a precise definition (with reference to the relevant distance or Hoffman-type constant) should appear in §2.
  2. [Numerical Experiments] Numerical section: the choice of the inconsistency parameter and the stopping tolerance for the inner alternating-projection loops should be stated explicitly to permit exact reproduction of the reported timings and iteration counts.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. The comments help clarify the presentation of the convergence results and the subproblem design. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [§4 (Convergence Analysis)] §4 (Convergence Analysis), main theorem on linear rates: the linear convergence of both radii and (unique-solution) iterates is asserted to follow from a 'natural error bound' between the constructed feasible set and the optimal ℓ1-ball, yet no explicit positive lower bound on this constant is derived in terms of A, b or the current radius. Without such a bound the claimed uniform linear rate is not guaranteed and the central convergence result remains conditional on an unverified assumption.

    Authors: The manuscript does not assert a uniform linear rate that is independent of the instance. Theorem 4.1 establishes linear convergence of the radii to the optimal value and, under uniqueness, linear convergence of the iterates, with the rate governed by the natural error-bound constant between each constructed feasible set and the optimal ℓ1-ball. This constant is strictly positive for any fixed feasible pair (A, b) possessing a unique solution, because the deliberate inconsistency in the subproblem construction ensures a positive separation that is preserved by the equivalence to basis pursuit. We will revise the statement of the main theorem and add a short remark after the proof to make explicit that positivity follows from uniqueness and the construction; we will also note that the resulting rate is necessarily instance-dependent. Deriving a closed-form, instance-independent lower bound on the constant is not feasible in general and is not claimed in the paper. revision: partial

  2. Referee: [§3 (Subproblem construction)] §3 (Subproblem construction): the manuscript states that the subproblems are designed so their feasible sets are disjoint 'by construction,' but does not supply a self-contained verification that this design preserves equivalence to the original basis-pursuit problem and does not inadvertently make the error-bound constant vanish for feasible (A,b) pairs.

    Authors: We agree that an explicit verification strengthens the exposition. In the revised manuscript we will insert a short proposition in §3 that (i) confirms the sequence of inconsistent subproblems is equivalent to the original basis-pursuit problem in the sense that the optimal value is recovered in the limit and (ii) shows that the enforced disjointness does not force the error-bound constant to zero for any feasible (A, b). The argument relies on the fact that the radii are monotonically nonincreasing and bounded below by the optimal ℓ1-norm, together with the uniqueness assumption guaranteeing a positive distance between the constructed sets and the optimal ball. revision: yes

standing simulated objections not resolved
  • Supplying an explicit instance-independent positive lower bound on the governing error-bound constant (the referee's request for a uniform rate guarantee across all instances).

Circularity Check

1 steps flagged

Linear convergence rate governed by self-cited natural error bound

specific steps
  1. self citation load bearing [Abstract]
    "Recently in [R. Behling, Y. Bello-Cruz and L.-R. Santos, SIAM J. Optim., 31 (2021), pp. 2863-2892], inconsistency coming from infeasibility has been shown to accelerate convergence of alternating projections. We deliberately enforce this inconsistency by constructing subproblems whose feasible sets are disjoint by design. We prove that the resulting ℓ1-radii converge linearly to the optimal value, and that when the solution is unique, all generated sequences converge linearly to it at a rate governed by a natural error bound between the feasible set and the optimal ℓ1-ball."

    The linear rate for both radii and sequences is explicitly stated to be governed by the natural error bound, whose acceleration property and applicability are justified solely by citation to the authors' overlapping prior work rather than by an independent derivation or explicit lower bound on the constant within this manuscript.

full rationale

The paper's central convergence result for ℓ1-radii and iterates invokes a natural error bound between the feasible set and optimal ℓ1-ball, with the inconsistency-acceleration property imported directly from the authors' own 2021 SIAM J. Optim. paper. This creates moderate dependence on prior self-work for the rate analysis, even though the specific subproblem construction for basis pursuit is new and the manuscript supplies its own proof structure around the bound. No self-definitional reductions, fitted predictions, or ansatz smuggling appear in the given text; the derivation remains partially independent.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard convex feasibility assumptions plus the existence of a natural error bound and the ability to construct disjoint subproblem sets; no free parameters or new entities are introduced.

axioms (2)
  • domain assumption A natural error bound exists between the feasible set and the optimal ℓ1-ball that controls the linear convergence rate.
    Invoked to govern the rate when the solution is unique.
  • domain assumption Subproblems can be constructed whose feasible sets are disjoint by design.
    Required to enforce the deliberate inconsistency that accelerates convergence.

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discussion (0)

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Reference graph

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