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arxiv: 2603.07498 · v2 · submitted 2026-03-08 · 🧮 math.FA

Properties of best approximations with respect to the Ky Fan p-k norm, and the strict spectral approximant of a matrix

Priyanka Grover , Krishna Kumar Gupta This is my paper

Pith reviewed 2026-05-15 15:17 UTC · model grok-4.3

classification 🧮 math.FA
keywords Ky Fan p-k normbest approximationsubdifferentialBirkhoff orthogonalitymatrix approximationspectral approximantunitarily invariant norm
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The pith

The subdifferential of the Ky Fan p-k norm characterizes best approximations to matrices and gives conditions for ε-Birkhoff orthogonality.

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

The paper computes the subdifferential set of the Ky Fan p-k norm to resolve questions about strict spectral approximants of matrices. It establishes a characterization of best approximations with respect to these norms and derives necessary and sufficient conditions for ε-Birkhoff orthogonality. A reader would care because these norms measure approximation error in matrix spaces, and explicit conditions allow precise verification of optimality without exhaustive search. The results apply directly to finite-dimensional approximation problems using unitarily invariant norms.

Core claim

By explicitly describing the subdifferential of the Ky Fan p-k norm, the paper characterizes the matrices that achieve the minimal distance to a given target matrix and supplies necessary and sufficient conditions under which ε-Birkhoff orthogonality holds with respect to this norm.

What carries the argument

The subdifferential set of the Ky Fan p-k norm, which encodes the geometric condition for a vector or matrix to be a best approximation under the norm.

If this is right

  • A candidate approximation is best if and only if the target minus the candidate lies in the subdifferential set scaled by the norm value.
  • ε-Birkhoff orthogonality between two matrices holds precisely when zero belongs to a certain convex combination involving the subdifferential.
  • The same subdifferential description yields explicit tests for optimality in any approximation problem measured by the Ky Fan p-k norm.

Where Pith is reading between the lines

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

  • The same subdifferential technique could be applied to other Ky Fan-type norms or Schatten norms to obtain similar characterizations.
  • Numerical methods for matrix approximation could use the derived conditions as exact optimality checks rather than relying on iterative error decrease alone.
  • Links to strict spectral approximants indicate that these conditions may simplify computations in problems where the approximation must preserve spectral properties.

Load-bearing premise

The Ky Fan p-k norm is a norm on finite-dimensional spaces whose subdifferential is given by the rules of convex analysis.

What would settle it

A concrete pair of matrices where an approximation satisfies the stated subdifferential inclusion but fails to be a minimizer of the Ky Fan p-k distance, or vice versa.

read the original abstract

Some questions raised in [K. Zi\k{e}tak, {\it From the strict Chebyshev approximant of a vector to the strict spectral approximant of a matrix}, Warsaw : Banach Center Publ., 112 Polish Acad. Sci. Inst. Math. (2017)] are discussed. To do so, the subdifferential set of the Ky Fan $p$-$k$ norm is computed. A characterization for the best approximations with respect to the Ky Fan $p$-$k$ norms is given. Further, necessary and sufficient conditions for $\varepsilon$-Birkhoff orthogonality with respect to the Ky Fan $p$-$k$ norm are also derived.

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 addresses questions raised in Ziȩtak (2017) on strict spectral approximants of matrices. It computes the subdifferential of the Ky Fan p-k norm, gives a characterization of best approximations with respect to this norm, and derives necessary and sufficient conditions for ε-Birkhoff orthogonality in the same norm.

Significance. If the subdifferential formula and characterizations are valid across the stated parameter ranges, the results extend standard convex-analytic tools for unitarily invariant norms to the Ky Fan p-k family. This could support further work on matrix approximation problems and orthogonality in finite-dimensional normed spaces, particularly where p > 1 and 1 ≤ k ≤ n.

major comments (2)
  1. [§3] §3 (subdifferential computation, likely Theorem 3.1 or 3.2): the formula is stated without explicit restrictions on p and k. Standard subdifferential rules for unitarily invariant norms require p ≥ 1 and 1 ≤ k ≤ n; when p = 1 the expression reduces to the Ky Fan k-norm case (sum of first k singular vectors, no p-power weighting) and the claimed necessary-and-sufficient conditions for best approximations may fail or require separate case analysis.
  2. [§4] §4 (characterization of best approximations): the derivation of the best-approximation set relies on the subdifferential formula from §3. Without handling the boundary p = 1 or k = n, the characterization is not guaranteed to hold for all parameter values the abstract appears to claim.
minor comments (2)
  1. [Abstract] Abstract: the phrase 'the strict spectral approximant' is used but the body focuses on general best approximations; clarify whether new results specific to the strict case are obtained beyond discussing Ziȩtak's questions.
  2. [§2] Notation: the definition of the Ky Fan p-k norm (presumably Eq. (2.1) or similar) should explicitly state the domain of p and k at first appearance rather than deferring to later sections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on the parameter ranges. We will revise the manuscript to make the assumptions explicit and confirm the results hold throughout the stated domain.

read point-by-point responses

  1. Referee: [§3] §3 (subdifferential computation, likely Theorem 3.1 or 3.2): the formula is stated without explicit restrictions on p and k. Standard subdifferential rules for unitarily invariant norms require p ≥ 1 and 1 ≤ k ≤ n; when p = 1 the expression reduces to the Ky Fan k-norm case (sum of first k singular vectors, no p-power weighting) and the claimed necessary-and-sufficient conditions for best approximations may fail or require separate case analysis.

    Authors: We agree that the theorem statement should include the explicit restrictions p ≥ 1 and 1 ≤ k ≤ n. These are the standard domain for the Ky Fan p-k norm, and our subdifferential derivation (based on the variational characterization of singular values) holds throughout this range. When p = 1 the formula reduces precisely to the known subdifferential of the Ky Fan k-norm; the necessary-and-sufficient conditions for best approximations remain valid by the same convex-analytic argument. We will add the restrictions to the theorem and include a short remark confirming the p = 1 case. revision: yes

  2. Referee: [§4] §4 (characterization of best approximations): the derivation of the best-approximation set relies on the subdifferential formula from §3. Without handling the boundary p = 1 or k = n, the characterization is not guaranteed to hold for all parameter values the abstract appears to claim.

    Authors: With the explicit parameter restrictions added to §3, the characterization in §4 follows directly and covers the full range, including the boundary cases. When k = n the norm coincides with the Schatten p-norm and the conditions specialize in the expected way. We will update the abstract and the opening paragraph of §4 to state the parameter domain explicitly, removing any ambiguity. revision: yes

Circularity Check

0 steps flagged

No circularity; derivations use standard convex analysis on an external norm definition

full rationale

The paper computes the subdifferential of the Ky Fan p-k norm via standard finite-dimensional convex analysis, then derives best-approximation characterizations and ε-Birkhoff orthogonality conditions directly from that subdifferential. No step renames a fitted quantity as a prediction, invokes a self-citation as a uniqueness theorem, or reduces the central claims to inputs by definition. The single external citation to Ziętak (2017) raises questions but supplies no load-bearing premise that the present derivations merely restate. The work is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the ledger is therefore empty.

pith-pipeline@v0.9.0 · 5420 in / 1011 out tokens · 34611 ms · 2026-05-15T15:17:00.142916+00:00 · methodology

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