arxiv: 2604.23104 · v1 · submitted 2026-04-25 · 🧮 math.NA · cs.NA· math.OC

Recognition: unknown

Rank One Completion for Higher Order Tensors

Linghao Zhang , Ioana Dumitriu , Jiawang Nie

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Pith reviewed 2026-05-08 07:52 UTC · model grok-4.3

classification 🧮 math.NA cs.NAmath.OC

keywords rank one tensor completionhigher order tensorsrecursive algorithmlinear systemssingular vectorstensor ranknoiseless completionnoisy data recovery

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The pith

A recursive algorithm uniquely completes rank one higher-order tensors using linear solves and singular vectors.

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

The paper defines rank one determinable tensors and develops a recursive algorithm for their completion. The algorithm reduces the problem to solving linear systems and extracting singular vectors at each step. In the absence of noise, it guarantees a unique rank one completion when the tensor meets the determinability conditions. With sufficiently small noise, the output remains close to the true completion. The approach is demonstrated to be efficient through numerical tests on various tensor orders.

Core claim

For tensors that are rank one determinable, the rank one completion can be computed recursively by solving linear systems and computing singular vectors, resulting in a unique solution without noise and a stable approximation when noise is small.

What carries the argument

The recursive algorithm that decomposes higher-order rank one completion into successive linear system solves and singular vector computations for lower dimensional problems.

If this is right

The completion is unique for noiseless rank one determinable tensors.
The method produces completions close to the true one when noise levels are low.
The algorithm applies to tensors of any order with only basic linear algebra operations.
Numerical results confirm efficiency and accuracy across tested cases.

Where Pith is reading between the lines

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

This could lead to extensions for completing tensors with other low-rank structures beyond rank one.
Similar recursive strategies might apply to related problems in multilinear algebra where uniqueness conditions can be established.
Practical implementations could integrate this into larger tensor processing pipelines for missing data recovery.

Load-bearing premise

The observed tensor entries must come from a rank one determinable tensor that allows the recursive procedure to determine a unique completion at each reduction step.

What would settle it

Finding a rank one determinable tensor with partial observations where two different rank one tensors match the observations but the algorithm selects one inconsistently would disprove uniqueness.

read the original abstract

We study the rank one completion problem for tensors of arbitrary orders. The notion of rank one determinable tensors is introduced. We explore its properties and propose a recursive algorithm for computing rank one tensor completion. This algorithm only requires solving linear systems and computing singular vectors. In the absence of noise, it produces a unique rank one completion under some assumptions. In the presence of noise, we show that the computed rank one tensor completion is close to the exact one when the noise is sufficiently small. Numerical experiments demonstrate the efficiency and accuracy of the proposed method.

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 recursive algorithm that completes higher-order tensors to rank one by solving linear systems and computing singular vectors, under an explicitly defined determinable condition.

read the letter

The core contribution is a constructive recursive procedure that turns rank-one tensor completion into a sequence of linear solves plus singular-vector steps. The authors introduce the notion of rank-one determinable tensors and prove that this property plus nonsingularity of the intermediate systems yields a unique completion when there is no noise. They also give a standard perturbation argument showing stability for small noise. Numerical experiments are included to check speed and accuracy on examples. That is the actual new piece: a tailored, implementable method rather than another optimization formulation. The approach is clean for numerical linear algebra work because it stays inside basic operations that scale reasonably with order. The assumptions are stated up front, which helps. The main limitation is that the determinable condition is restrictive; many partially observed tensors will fail the nonsingularity requirements at some recursion step, so the method applies only to a subset of instances. It is not clear from the abstract how large that subset is in practice or how the failure rate behaves with order and sampling density. The stability result is the usual first-order bound, so it does not claim robustness beyond small noise. No circularity appears in the argument structure. This is useful reading for researchers who already work on structured tensor recovery or low-rank completion in numerical analysis. A reader who needs an explicit, non-iterative procedure for exact rank-one cases will find the algorithm and the supporting theory worth examining. The paper is solid enough on its own terms to go to referees; the proofs and the practical range of the assumptions are the natural points for detailed checking.

Referee Report

2 major / 3 minor

Summary. The manuscript introduces the notion of rank-one determinable tensors and develops a recursive algorithm for rank-one completion of higher-order tensors. The algorithm reduces the problem to a sequence of linear system solves and singular-vector computations. It claims that, for rank-one determinable inputs satisfying the stated nonsingularity conditions, the procedure yields a unique completion in the noiseless case and a stable approximation when the noise level is sufficiently small, with supporting numerical experiments.

Significance. If the claims hold, the work supplies a constructive, computationally lightweight method for a core tensor-completion task that appears in multilinear algebra and data-analysis applications. The reduction to standard linear-algebra primitives and the explicit conditioning on the rank-one-determinable property constitute clear algorithmic strengths.

major comments (2)

[Section 3] The uniqueness statement in the noiseless case is conditional on the input being rank-one determinable and on the associated linear systems being nonsingular; the manuscript should supply a precise, checkable criterion (beyond the recursive definition) that guarantees these systems remain nonsingular for tensors of arbitrary order.
[Section 4] The stability result is presented as a standard perturbation argument for the linear solves; an explicit bound relating the noise level to the condition numbers of the successive linear systems would strengthen the claim and clarify its dependence on the tensor order.

minor comments (3)

[Abstract] The abstract refers to 'some assumptions' without naming the rank-one-determinable property; a single sentence linking the two would improve readability.
[Section 2] Notation for the recursive index sets and the unfolding operators should be introduced once and used consistently; occasional redefinition in later sections obscures the flow.
[Section 5] Numerical experiments would benefit from a brief statement of the condition-number ranges observed across the test tensors, to illustrate the practical scope of the stability result.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the insightful comments, which help clarify the presentation of our results. We address each major comment below.

read point-by-point responses

Referee: [Section 3] The uniqueness statement in the noiseless case is conditional on the input being rank-one determinable and on the associated linear systems being nonsingular; the manuscript should supply a precise, checkable criterion (beyond the recursive definition) that guarantees these systems remain nonsingular for tensors of arbitrary order.

Authors: The rank-one determinable property is defined recursively because the completion procedure itself proceeds recursively by reducing the order of the tensor at each step. Consequently, the nonsingularity conditions on the successive linear systems are also verified sequentially during algorithm execution; each condition is directly checkable by inspecting the matrices constructed from the input data at that stage. For tensors of arbitrary order a non-recursive, closed-form criterion is not available in general, as the conditions depend on the specific unfolding and the preceding singular-vector computations. We will add a remark in Section 3 clarifying that the recursive definition furnishes a practical, computationally verifiable criterion that can be checked while running the algorithm, without requiring any additional symbolic analysis. revision: partial
Referee: [Section 4] The stability result is presented as a standard perturbation argument for the linear solves; an explicit bound relating the noise level to the condition numbers of the successive linear systems would strengthen the claim and clarify its dependence on the tensor order.

Authors: We agree that an explicit perturbation bound would make the stability statement more precise. In the revised manuscript we will derive such a bound by applying standard linear-algebra perturbation theory to each recursive step, explicitly tracking the condition numbers of the linear systems and the growth with tensor order. The resulting estimate will be stated in Section 4 together with a brief discussion of its dependence on the order. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces the rank-one-determinable property as an explicit assumption on the input tensor, then constructs a recursive completion procedure that reduces to a sequence of linear-system solves and singular-vector extractions. Uniqueness holds precisely when those systems are nonsingular and the determinable condition is met; both are stated independently of the output tensor. The noise-stability claim is the standard first-order perturbation bound on the same linear operations. No equation equates the claimed completion to a fitted or self-referential quantity, and no load-bearing step collapses to a prior self-citation or ansatz that presupposes the result.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claims rest on a newly introduced definition and on domain assumptions about uniqueness that are not independently verified outside the paper.

axioms (1)

domain assumption The tensor belongs to the class of rank one determinable tensors
This property is required for the uniqueness statement and is defined within the paper.

invented entities (1)

rank one determinable tensor no independent evidence
purpose: Characterizes tensors for which a unique rank-one completion exists from partial observations
New definition introduced to support the algorithm and its guarantees; no independent evidence supplied.

pith-pipeline@v0.9.0 · 5384 in / 1249 out tokens · 97342 ms · 2026-05-08T07:52:24.194637+00:00 · methodology

discussion (0)

Reference graph

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