Minimum Variance Solution of Underdetermined Systems of Linear Equations

Davide Elia; Lorenzo Piazzo; Sergio Molinari

arxiv: 1906.09121 · v1 · pith:DTAHL2ILnew · submitted 2019-06-21 · 🧮 math.OC · eess.SP

Minimum Variance Solution of Underdetermined Systems of Linear Equations

Lorenzo Piazzo , Davide Elia , Sergio Molinari This is my paper

Pith reviewed 2026-05-25 18:38 UTC · model grok-4.3

classification 🧮 math.OC eess.SP

keywords underdetermined linear systemsminimum variance solutionminimum norm solutionclosed form expressionlinear equationspseudoinverse

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The pith

Underdetermined linear systems have a Minimum Variance solution with a simple closed-form expression.

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

The paper examines underdetermined systems of linear equations, which admit infinitely many solutions. It introduces the Minimum Variance solution as a distinct alternative to the Minimum Norm solution. The authors derive its properties and present an explicit closed-form formula for computing it. A reader would care because many inverse problems and fitting tasks require picking one solution from the infinite set, and variance minimization offers a statistically motivated choice.

Core claim

For the underdetermined system Ax = b, the Minimum Variance solution is the particular solution that minimizes the variance of the entries in x, and this solution possesses a closed-form expression that can be written down directly once the system matrix is known.

What carries the argument

The Minimum Variance solution, obtained by minimizing the variance of x subject to the linear constraint Ax = b and shown to admit an explicit algebraic expression related to but separate from the pseudoinverse formula for the Minimum Norm solution.

If this is right

The MV solution can be evaluated by direct matrix operations without iterative numerical optimization.
Properties such as uniqueness or stability of the MV solution follow immediately from the closed-form expression.
The MV solution reduces to the MN solution only in special cases, allowing systematic comparison between the two.
Any application that already uses the MN solution can substitute the MV formula with no change in computational complexity.

Where Pith is reading between the lines

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

The same algebraic device that yields the MV formula may apply to other quadratic criteria on the solution vector.
In statistical estimation settings the MV solution could serve as a baseline for comparing regularized estimators.
Extensions to weighted variance or block-structured systems would follow the same derivation steps.

Load-bearing premise

That a distinct and useful Minimum Variance solution exists whose properties can be derived in closed form independently of additional modeling choices such as weighting or regularization parameters.

What would settle it

An explicit underdetermined system Ax = b for which no closed-form expression for the variance-minimizing solution exists, or for which that solution is identical to the minimum-norm solution for every right-hand side b.

read the original abstract

A system of linear equations is said underdetermined when there are more unknowns than equations. Such systems may have infinitely many solutions. In this case, it is important to single out solutions possessing special features. A well known example is the Minimum Norm (MN) solution, which is the solution having the least Euclidean norm. In this note, we consider another useful solution, related with the MN one, which we call the Minimum Variance (MV) solution. We discuss some of its properties and derive a simple, closed form expression.

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.

Desk Editor's Note private letter to a colleague

The note defines a minimum-variance solution for underdetermined Ax=b and claims a simple closed form distinct from the minimum-norm pseudoinverse solution.

read the letter

The main takeaway is that the authors introduce a Minimum Variance (MV) solution for underdetermined linear systems as an alternative to the standard minimum-norm one and say they have derived a closed-form expression for it. They discuss a few properties along the way. That is the entire contribution of the note. If the derivation is correct and the MV solution is genuinely different without extra parameters, it gives practitioners another explicit way to pick one solution out of the affine space. The paper does that job cleanly on its own terms: it stays within basic linear algebra, avoids unnecessary regularization, and keeps the claim modest. The abstract and stress-test give no sign of circularity or hidden fitting, so the central claim appears to rest on standard identities applied in a new labeling. The soft spot is simply that we only have the abstract here. Without the actual equations it is impossible to check whether the closed form is new, whether it reduces to a weighted pseudoinverse under the hood, or whether the variance interpretation requires an implicit covariance matrix that the authors have not declared. Those are the only points that need verification; nothing else looks load-bearing. This is a short technical note aimed at people who routinely solve underdetermined systems in optimization, signal processing, or statistics and want an explicit alternative selection rule. A reader already comfortable with the Moore-Penrose pseudoinverse will get the value immediately if the algebra checks. It is not a major reorganization of the field, but the claim is narrow enough and the setting standard enough that it deserves a serious referee rather than a desk rejection. I would send it out.

Referee Report

0 major / 2 minor

Summary. The manuscript introduces the Minimum Variance (MV) solution for underdetermined linear systems Ax = b as a useful alternative to the well-known Minimum Norm (MN) solution obtained via the Moore-Penrose pseudoinverse. It discusses properties of the MV solution and derives a simple closed-form expression for it.

Significance. If the claimed closed-form expression is correct and the MV solution is mathematically distinct from the MN solution without introducing auxiliary weighting or covariance matrices, the result would supply a parameter-free selection criterion for solutions of underdetermined systems. This could be of interest in optimization and statistics contexts where variance minimization is natural, and the provision of an explicit formula would be a practical contribution.

minor comments (2)

The abstract and introduction should explicitly state the precise definition of 'variance' used for the MV solution (e.g., whether it assumes an identity covariance or derives from a specific quadratic form) to avoid ambiguity with standard MN properties.
Any numerical examples or verification against the MN solution should include the explicit matrix A and vector b to allow readers to reproduce the claimed distinction between MV and MN solutions.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of the manuscript and the recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces the Minimum Variance solution for underdetermined linear systems as a concept related to but distinct from the standard Minimum Norm (pseudoinverse) solution, then derives its closed-form expression. The abstract and description supply no equations, self-citations, fitted parameters, or ansatzes that reduce the claimed result to its inputs by construction. The derivation is presented as resting on ordinary linear-algebra identities without load-bearing self-references or redefinitions, making the central claim self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, axioms, or invented entities are visible. The derivation is described as algebraic and therefore presumed to rest on standard linear-algebra facts.

pith-pipeline@v0.9.0 · 5609 in / 985 out tokens · 20817 ms · 2026-05-25T18:38:33.829906+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

[1]

W. R. Madych, ”Solutions of Underdetermined Sys- tems of Linear Equations”, Lecture Notes-Monograph Series Vol. 20, Spatial Statistics and Imaging, pp. 227- 238, 1991

work page 1991
[2]

C. D. Meyer: ”Matrix analysis and applied linear al- gebra”, SIAM, 2000

work page 2000
[3]

C. L. Lawson and R. J. Hanson: ”Solving Least Squares problems”, SIAM, 1995. 3

work page 1995

[1] [1]

W. R. Madych, ”Solutions of Underdetermined Sys- tems of Linear Equations”, Lecture Notes-Monograph Series Vol. 20, Spatial Statistics and Imaging, pp. 227- 238, 1991

work page 1991

[2] [2]

C. D. Meyer: ”Matrix analysis and applied linear al- gebra”, SIAM, 2000

work page 2000

[3] [3]

C. L. Lawson and R. J. Hanson: ”Solving Least Squares problems”, SIAM, 1995. 3

work page 1995