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Identifiability through special linear measurements

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Pith's one-line read The paper proves that dim X + 1 generic measurements from any irreducible, non-degenerate family L of linear functionals uniquely identify any point of an algebraic variety X, and this bound cannot be improved in general.

desk verdict A correct and genuinely useful generalization of Noether Normalization to structured linear measurements; the main proof holds up, with only minor exposition issues. read the letter →

arxiv 2505.24328 v1 pith:3HUY4JBX submitted 2025-05-30 math.AG cs.ITcs.NAmath.ITmath.NA

Fulvio Gesmundo , Alexandros Grosdos , André Uschmajew This is my paper
classification math.AGcs.ITcs.NAmath.ITmath.NA MSC 14Q1515A2990C30
keywords linearmeasurementsalgebraicvarietiesidentifiabilityNoethernormalizationlow-rankmatrixrecoverycompressedsensingpointevaluationsprojectivegeometry
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper establishes a general identifiability threshold for recovering a point on an algebraic variety from linear measurements that are themselves constrained to a variety. The result is: if X is any algebraic variety in a finite-dimensional vector space and L is any irreducible algebraic variety of measurement functionals not contained in one hyperplane, then $\dim X + 1$ generic measurements chosen from L determine every point $x\in X$ uniquely. The theorem holds over every infinite field and covers settings where both the model and the measurements are nonlinear, such as low-rank matrices with rank-one measurements or polynomials from point evaluations. The bound is sharp: in typical cases, $\dim X$ generic measurements leave several candidates, as shown by a parabola in the plane and by generic plane curves of degree at least four.

What carries the argument

The load-bearing object is the pair of statements Lemma 2.1 and Proposition 2.2. Lemma 2.1 says that because $L$ is irreducible and not contained in any hyperplane, for any two distinct vectors $v_1,v_2$ the set of $\ell\in L$ with $\ell(v_1)=\ell(v_2)$ is a proper subvariety of $L$; hence a generic measurement separates any fixed pair. Proposition 2.2 iterates this: a generic first measurement strictly lowers the dimension of every positive-dimensional irreducible component of $X$ on the affine hyperplane where it matches $x$, so after $n=\dim X$ cuts the candidate set is finite, and the extra measurement supplied by Lemma 2.1 eliminates all candidates but $x$. This is a restricted version of a Noether-normalization slicing argument, with all slices drawn from $L$ rather than from all of $V^*$.

What would settle it

A concrete way to test the claim is to search for an irreducible algebraic variety $L\subseteq V^*$ not contained in a hyperplane and a variety $X$ with a point $x$ such that every tuple of $\dim X+1$ elements of $L$ in some Zariski-open subset leaves at least two points of $X$ with identical measurement values; the existence of such a pair would falsify Theorem 1.1. A tractable first check would be small cases such as conics and cubic plane curves with $L$ the Veronese surface of point evaluations.

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Extended reading notes

Core claim

The central claim, Theorem 1.1, is that for a finite-dimensional vector space $V$, an algebraic variety $X\subseteq V$, a point $x\in X$, and an irreducible algebraic variety $L\subseteq V^*$ not contained in any hyperplane, generic elements $\ell_0,\ldots,\ell_n\in L$ with $n=\dim X$ uniquely identify $x$ from the measurements $\ell_i(x)$. The proof works by cutting: $n$ generic measurements restrict $X$ to a finite set (Proposition 2.2), and one further measurement separates the finitely many remaining candidates (Lemma 2.1). The projective formulation (Theorem 2.3) says that the rational projection from a generic tuple of $n+2$ elements of $L$ is injective on $X$. The paper also shows that $n+1$ is optimal in general: for a parabola a generic single measurement leaves two points, and for a generic plane curve of degree at least four no point is determined by a single measurement.

Load-bearing premise

The measurement family $L$ must be an irreducible algebraic variety and must not lie inside any single hyperplane of $V^*$, so that any two distinct candidate points are separated by a Zariski-open set of measurements.

Editorial extensions

If this is right

  • For rank-one (bilinear) measurements on the variety of rank-at-most-$k$ matrices, Theorem 1.1 guarantees that $\dim X + 1 = (d_1+d_2-k)k + 1$ generic measurements identify a rank-$k$ matrix uniquely.
  • For polynomial recovery from point evaluations, any polynomial lying in a variety model of dimension $n$—sparse polynomials, low Waring rank, structured circuits—is identified by $n+1$ generic evaluations.
  • For tensor-network and feature-map models used in learning, the theorem gives a sharp sample complexity for noiseless exact identification with generic rank-one samples.
  • In the projective setting, a generic projection from $L^{n+2}$ is injective on $X$, so the result can be read as a Noether-normalization-type statement for restricted projections.
  • The bound cannot be improved in general: the examples show that $\dim X$ generic measurements are not enough even for simple curves, so the extra measurement is not an artifact of the proof.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper leaves open quantitative versions: how large are the Zariski-open sets of good measurements, and can failure probabilities be bounded for specific pairs $(X,L)$; the recursive construction suggests such bounds could be derived for structured families.
  • Irreducibility of $L$ is used only to make the separating set proper, so a finite union of irreducible non-degenerate components might behave similarly if the components are jointly non-degenerate, but the paper does not claim this.
  • The theorem fails for finite measurement sets such as single-entry matrix completion, so a discrete analogue would need additional combinatorial or incoherence assumptions; the gap between continuous irreducible families and finite sampling sets deserves separate study.
  • For generic plane curves the exceptional points recoverable with $\dim X$ measurements are finite, and the paper sketches that for higher-dimensional varieties one might expect exceptional loci of positive dimension; this is an extrapolation beyond the curve case.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 6 minor

Summary. The paper studies the minimum number of generic linear measurements, chosen from a prescribed irreducible algebraic family L ⊆ V*, needed to uniquely identify a point x on an algebraic variety X ⊆ V. The main result (Theorem 1.1) states that if L is not contained in any hyperplane of V*, then dim X + 1 generic measurements suffice. The proof is an induction on dimension: Lemma 2.1 shows that a generic element of L separates any fixed pair of distinct points, and Proposition 2.2 shows that n generic hyperplane sections cut an n-dimensional variety down to a finite set; a final generic measurement then separates the finite candidates. A projective analogue (Theorem 2.3) and examples illustrating sharpness (a cubic curve requiring dim X + 1 measurements) are also given.

Significance. The result is a clean, broadly applicable identifiability statement that generalizes the Noether Normalization Lemma to structured measurement families. It applies to bilinear measurements, point evaluations of polynomials, and tensor-network feature maps. The proof is self-contained, short, and uses only standard facts from algebraic geometry. The sharpness example is explicit and convincing. If the minor issues below are fixed, the paper will be a useful reference for algebraic compressed sensing and related areas.

minor comments (6)
  1. [Section 2.2, Lemma 2.1] The lemma should explicitly assume v1 ≠ v2; otherwise H = V* and the conclusion that C is strictly contained in L is false.
  2. [Section 2.2, Proposition 2.2] The step "By Lemma 2.1, a generic element ℓ ∈ L is non-constant on every X^(i)" is not a direct consequence of Lemma 2.1, since the latter applies to a fixed pair (v1,v2). The intended argument is that the set of linear forms constant on a positive-dimensional irreducible component is a proper linear subspace W of V*, and L ∩ W is a proper closed subset because L is not contained in any hyperplane; this should be stated explicitly.
  3. [Section 2.2, proof of Proposition 2.2] There is a typo: "ξ1" should be "y1" in the definition of X1.
  4. [Section 2.3, Theorem 2.3 and its proof] The coordinate description of the map should be [v] ↦ [ℓ0(v):...:ℓ_{n+1}(v)] into P^{n+1} (or the appropriate projective space of the quotient), and the well-definedness argument should conclude ⟨ℓ0,...,ℓ_{n+1}⟩^⊥ ∩ X = ∅ rather than ⟨ℓ0,...,ℓ_n⟩^⊥ ∩ X = ∅.
  5. [Section 2.3, proof of Theorem 2.3] After dehomogenizing by ℓ_{n+1}=1, the proof should note that the restricted measurement variety remains irreducible and is not contained in a hyperplane of the affine space, so that Theorem 1.1 applies.
  6. [Section 1.1, remark after Theorem 1.1] The statement "Theorem 1.1 holds over every infinite field" should be qualified, since for a general infinite field a nonempty Zariski-open subset of L may contain no k-rational points; the standard interpretation over R and C is clear.

Circularity Check

0 steps flagged · score 0.0 of 10

Derivation is self-contained: Theorem 1.1 follows from standard algebraic geometry facts, with no fitted parameters, no load-bearing self-citations, and no definitional circularity.

full rationale

The proof of Theorem 1.1 is fully self-contained and does not assume the target result. Lemma 2.1 is an immediate consequence of the hypotheses on L (irreducible and not contained in any hyperplane): the set C = {ℓ ∈ L : ℓ(v1) = ℓ(v2)} is L ∩ H for a hyperplane H, hence a proper subvariety. Proposition 2.2 then proceeds by a direct induction on dim X: a generic ℓ ∈ L is non-constant on each positive-dimensional irreducible component X^(i), so the hyperplane section X^(i) ∩ H(ℓ − y) has strictly smaller dimension, and after n = dim X cuts the candidate set is finite. Theorem 1.1 combines these two ingredients: after n cuts the set X0 is finite, and the final generic measurement ℓ0 separates the finitely many candidates by Lemma 2.1. No parameter is fitted, no quantity is defined in terms of the conclusion, and the theorem is not an input to its own proof. The only self-citations ([BGMV23] for generic measurements in matrix sensing and for finite measurement sets; [GHIL16] for algebro-matroid connections) appear in the introduction and Section 3.1 as context, and the paper explicitly disclaims their role: 'This discussion is not directly related to the proof of Theorem 1.1, but serves as a motivation for this work.' Two flagged weaknesses are minor and non-circular: Lemma 2.1 omits the hypothesis v1 ≠ v2 (otherwise H is not a hyperplane), a condition satisfied in every actual use; and the sharpness claim for curves of degree ≥ 4 in Section 3.3 is explicitly presented as an incomplete argument ('We omit the full proof and we only sketch the argument'), which concerns peripheral minimality, not the main derivation. The theorem therefore stands on its own, with the sharpness examples (parabola, conic, cubic curve) providing independent content showing the dim X + 1 bound cannot be improved in general.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The proof uses only standard algebraic geometry facts: dimension theory, irreducibility, genericity. No free parameters are introduced. The only structural assumption beyond basic setup is the irreducibility and linear non-degeneracy of the measurement variety L, which is an explicit hypothesis of the theorem.

assumptions (6)
  • standard math A proper subvariety of an irreducible variety of dimension m has dimension at most m-1 (Shafarevich, Theorem 1.19).
    Used in Proposition 2.2 to ensure each hyperplane cut strictly reduces dimension on irreducible components.
  • standard math Algebraic varieties have finitely many irreducible components with well-defined dimensions.
    Used in the induction in Proposition 2.2 to decompose X into irreducible components.
  • standard math For infinite fields, Zariski-open subsets are dense and non-empty; over R/C, proper closed subsets have measure zero.
    Used to interpret 'generic' and to connect the genericity statements with probability-one guarantees.
  • standard math Bezout's theorem and intersection theory in the plane (Harris, Chapter 18).
    Used in Section 3.3 to count intersections of lines with the cubic curve and to bound the dual curve degree.
  • standard math Properties of the dual curve of a smooth plane curve (Gelfand-Kapranov-Zelevinsky, Proposition 1.2.4).
    Used in the sketch for curves of degree at least four to assert that singularities of the dual curve are simple nodes or cusps.
  • domain assumption The measurement family L is irreducible and not contained in any hyperplane of V*.
    This is an explicit hypothesis of Theorem 1.1; it is the minimal condition under which Lemma 2.1 holds, and the whole construction depends on it.

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Pith. "Pith review of Identifiability through special linear measurements." pith.science (2026). https://pith.science/paper/3HUY4JBX

@misc{pith2026250524328,
  author       = {Pith},
  title        = {Pith review of: Identifiability through special linear measurements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3HUY4JBX}},
  note         = {Machine review of arXiv:2505.24328}
}
abstract

We show that one can always identify a point on an algebraic variety $X$ uniquely with $\dim X +1$ generic linear measurements taken themselves from a variety under minimal assumptions. As illustrated by several examples the result is sharp, that is, $\dim X$ measurements are in general not enough for unique identifiability.

Figures

Figures reproduced from arXiv: 2505.24328 by the authors.

Figure 1
Figure 1. The real part of a cubic curve with λ = 2. The red points p1, p2 are two of the nine inflection points. The tangent lines at the three blue points q1, q2, q3 are parallel and they meet at the point p∞. p∞ and p∞ is an inflection point for X. There are eight more inflection points p1, . . . , p8 on X, given by the eight solutions of the polynomial system 0 = x 2 2 − x1(x1 − 1)(x1 − λ); 0 = (λ(λ + 1 − 3x1)x1) + (λ − (… view at source ↗

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