On linear regression in three-dimensional Euclidean space

O. V. Ageev; R. A. Sharipov

arxiv: 1907.06009 · v1 · pith:SBWR2GLWnew · submitted 2019-07-13 · 💻 cs.CG

On linear regression in three-dimensional Euclidean space

O. V. Ageev , R. A. Sharipov This is my paper

Pith reviewed 2026-05-24 22:10 UTC · model grok-4.3

classification 💻 cs.CG

keywords linear regressionthree-dimensional Euclidean spacecoordinate-free methodbest-fit linespatial straight lineleast-squares fittingvector geometry

0 comments

The pith

The best-fit straight line to points in three-dimensional space admits a unique coordinate-free description.

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

The paper treats the task of locating the spatial line that minimizes the sum of squared distances to a given collection of points. It supplies an explicit solution written entirely without reference to any coordinate basis or origin choice. A reader would value this because coordinate choices are arbitrary and can obscure the intrinsic geometry of the fit. The resulting expression uses only vector operations that remain unchanged under rigid motions of the whole configuration.

Core claim

The three-dimensional linear regression problem of finding the straight line best fitting a finite set of points is solved by deriving the line's position and direction through coordinate-free geometric operations.

What carries the argument

Coordinate-free characterization of the regression line that expresses its location via the centroid of the points and its direction via an extremal vector that is independent of any basis.

If this is right

The best-fit line can be computed directly from vector averages and inner products without selecting an origin or axes.
The same geometric construction remains valid under any rigid motion of the entire point set.
The solution extends immediately to the case of weighted points by replacing ordinary averages with weighted ones.
No auxiliary coordinate transformations are required before or after the computation.

Where Pith is reading between the lines

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

The same intrinsic construction might be applied to regression problems on other Euclidean manifolds once an appropriate notion of straight line is fixed.
The method supplies a geometric route to the principal axis of a point cloud that bypasses matrix diagonalization.
Explicit formulas derived this way could be used to compare regression quality across different ambient dimensions without basis-dependent artifacts.

Load-bearing premise

A unique best-fit line exists for any finite collection of points and can be identified using only intrinsic geometric quantities without choosing coordinates.

What would settle it

A concrete finite set of points together with an explicit coordinate-free formula whose resulting line fails to achieve the global minimum of the sum of squared perpendicular distances when evaluated in any coordinate system.

read the original abstract

The three-dimensional linear regression problem is a problem of finding a spacial straight line best fitting a group of points in three-dimensional Euclidean space. This problem is considered in the present paper and a solution to it is given in a coordinate-free form.

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 paper gives a coordinate-free version of 3D line fitting but leaves the non-unique case when two largest eigenvalues match unaddressed.

read the letter

The main point is that the authors worked out an intrinsic, coordinate-free expression for the line that best fits a set of points in 3D by minimizing the sum of squared perpendicular distances. They express the solution using vectors and operations that do not depend on a chosen basis, which is the stated contribution. This is mostly a geometric rephrasing of the usual principal-component approach, with the line running through the centroid along the dominant direction of the covariance. That formulation can be convenient in settings where coordinate choices complicate proofs or code. The derivation itself appears straightforward and internally consistent for the generic case. The soft spot is the degeneracy. When the two largest eigenvalues of the covariance are equal, every direction in the corresponding plane achieves the same minimum error, so the solution is a whole pencil of lines rather than one. The abstract gives no sign that the paper outputs the full eigenspace or restricts to distinct eigenvalues, and the stress-test concern holds up on the given material. Without that clarification the characterization is incomplete. There are no examples, error analysis, or comparisons to coordinate methods, so the practical difference is hard to judge. The work is a narrow reformulation of a classical problem rather than a new result with fresh consequences. It is aimed at a small audience interested in intrinsic methods in computational geometry. Most readers will not find it useful. I would not bring it to a reading group, would not cite it, and would not send it for peer review because the novelty is limited and the gap on uniqueness is noticeable.

Referee Report

2 major / 0 minor

Summary. The manuscript addresses the three-dimensional linear regression problem of finding a spatial straight line that best fits a given group of points in Euclidean 3-space and asserts that a solution is supplied in coordinate-free form.

Significance. A rigorously derived coordinate-free characterization of the best-fit line (e.g., via intrinsic operations on the point set without reference to a basis) would be of interest in computational geometry for applications that require invariance under rigid motions. However, the absence of any explicit construction, proof, or verification steps prevents assessment of whether the claimed result holds or offers advantages over the standard principal-component approach.

major comments (2)

Abstract: the claim that 'a solution to it is given in a coordinate-free form' is unsupported; the manuscript contains no equations, derivation, error analysis, or explicit geometric construction that would allow verification or reproduction of the asserted solution.
Abstract (implicit uniqueness claim): when the two largest eigenvalues of the centered point covariance matrix coincide, the minimizer of the sum of squared perpendicular distances is any line through the centroid whose direction lies in the corresponding eigenplane; the manuscript gives no indication whether the coordinate-free solution outputs the full pencil or restricts to the generic (unique) case.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the comments. We agree that the submitted manuscript is extremely brief and does not contain the derivation or details needed to support the abstract claim. We will revise the manuscript to include the full coordinate-free derivation, geometric construction, and discussion of special cases.

read point-by-point responses

Referee: Abstract: the claim that 'a solution to it is given in a coordinate-free form' is unsupported; the manuscript contains no equations, derivation, error analysis, or explicit geometric construction that would allow verification or reproduction of the asserted solution.

Authors: We acknowledge that the current manuscript consists only of the abstract and provides no equations, derivation, or construction. This is a genuine omission. In the revised version we will supply an explicit coordinate-free expression for the minimizing line (using only the centroid and the principal direction obtained via intrinsic operations on the point set), together with its derivation, verification on example data, and comparison to the standard PCA approach. revision: yes
Referee: Abstract (implicit uniqueness claim): when the two largest eigenvalues of the centered point covariance matrix coincide, the minimizer of the sum of squared perpendicular distances is any line through the centroid whose direction lies in the corresponding eigenplane; the manuscript gives no indication whether the coordinate-free solution outputs the full pencil or restricts to the generic (unique) case.

Authors: The manuscript does not address the degenerate case. Our intended coordinate-free solution is formulated for the generic situation in which the two largest eigenvalues are distinct, yielding a unique direction. We will revise the text to state this limitation explicitly and to describe the solution set in the equal-eigenvalue case as the pencil of all lines through the centroid lying in the corresponding eigenplane. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation is a direct coordinate-free formulation of standard PCA

full rationale

The paper states it solves the 3D best-fit line problem in coordinate-free form. No equations, parameters, or citations appear in the abstract or description that reduce any claimed result to a fitted input, self-definition, or self-citation chain. The central claim is a mathematical re-expression of the known variance-maximization characterization, which is independent of the paper's own outputs and does not rely on prior author work for uniqueness or ansatz. This is the normal case of a self-contained geometric derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities can be identified from the given text.

pith-pipeline@v0.9.0 · 5551 in / 925 out tokens · 19586 ms · 2026-05-24T22:10:53.369049+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 4.1. A line is an optimal root mean square line ... if its direction vector a is directed along the primary axis of the non-linearity form Q ... corresponding to its minimal eigenvalue.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.