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arxiv: 2506.17463 · v3 · submitted 2025-06-20 · 🧮 math.ST · stat.ME· stat.TH

Testing Separability of High-Dimensional Covariance Matrices

Bongjung Sung , Peter D. Hoff This is my paper

Pith reviewed 2026-05-19 07:53 UTC · model grok-4.3

classification 🧮 math.ST stat.MEstat.TH
keywords separability testinghigh-dimensional covariancecore covariance matrixsphericity testmatrix-variate datainvariant testsasymptotic spectral equivalence
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The pith

Testing separability of a high-dimensional covariance matrix is equivalent to testing sphericity of its core component.

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

The paper shows that checking whether the covariance matrix Σ for matrix-variate data is separable is the same as checking whether its core component is spherical. This equivalence produces test statistics that remain well-defined when the dimension is large relative to sample size and whose distributions under the null do not depend on unknown parameters. Researchers can therefore simulate the null distribution exactly and control error rates precisely. The authors also prove that the sample covariance and its core are asymptotically equivalent in spectrum when the ratio of dimension to sample size converges to a positive constant, which justifies the use of these tests in the high-dimensional regime. Numerical comparisons indicate that the resulting procedures detect non-separability more often than earlier methods.

Core claim

We show that testing separability of Σ is equivalent to testing sphericity of its core component. With this insight, we construct test statistics that are well-defined in high-dimensional settings and have distributions that are invariant under the null hypothesis of separability, allowing for exact simulation of null distributions. We establish the asymptotic properties of some test statistics by proving the asymptotic spectral equivalence between the sample covariance matrix and its core in a p/n→γ∈(0,∞) regime.

What carries the argument

The core covariance matrix, a complementary object to a separable covariance matrix, carries the argument by converting separability testing into an equivalent sphericity test whose null distribution is parameter-free.

If this is right

  • Test statistics become invariant under the null, so their critical values can be obtained by direct simulation without estimating nuisance parameters.
  • The tests remain valid when the ratio of variables to observations converges to any positive finite limit.
  • The same equivalence supplies a route to other high-dimensional tests by replacing separability with other structural hypotheses on the core.
  • Numerical evidence indicates these procedures detect departures from separability more reliably than prior methods for the same sample sizes.

Where Pith is reading between the lines

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

  • The reduction to a sphericity problem on the core may simplify the derivation of tests for related covariance structures such as Kronecker or factor models.
  • Applied researchers working with matrix-valued observations could use the exact null simulation to set precise significance levels in settings where parameter-dependent approximations have been unreliable.
  • Finite-sample refinements that exploit the exact invariance might further improve performance when the dimension-to-sample ratio is moderate rather than large.

Load-bearing premise

The core covariance matrix is defined so that separability of the original matrix holds exactly if and only if the core is spherical, and the asymptotic spectral equivalence between sample and population versions is valid when dimension over sample size approaches a positive constant.

What would settle it

A Monte Carlo experiment in which the proposed test statistic under the null of separability produces a distribution that still depends on unknown parameters, or in which the tests do not show higher power than existing procedures, would falsify the central claims.

Figures

Figures reproduced from arXiv: 2506.17463 by Bongjung Sung, Peter D. Hoff.

Figure 1
Figure 1. Figure 1: Empirical power of the separability tests under Np1×p2 (0, C1,w) as functions of n across each w = 0.2, 0.4, 0.6, 0.8, and ˆγ := ˆγ1γˆ2 = p1p2/n. The tests ϕ1, ϕ2, and ϕ3 are those proposed in this article. LRT and PTCLT denote the tests by [41, 34] and [3], respectively. LRT is not present when ˆγ > 1, since its test statistic is not well-defined in this case [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Empirical power of the separability tests under Np1×p2 (0, C2,w) as functions of n across each w = 0.2, 0.4, 0.6, 0.8, and ˆγ := ˆγ1γˆ2 = p1p2/n. For the notations of ϕ1, ϕ2, ϕ3, LRT, and PTCLT, see [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Empirical power of the separability tests under Np1×p2 (0, C1,w) as functions of w across (ˆγ1, γˆ2). Here n = 256 and (p1, p2) is determined according to the value of (ˆγ1, γˆ2). For the notations of ϕ1, ϕ2, ϕ3, LRT, and PTCLT, see [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Empirical power of the separability tests under Np1×p2 (0, C1,w) as functions of w across (ˆγ1, γˆ2). Here n = 256 and (p1, p2) is determined according to the value of (ˆγ1, γˆ2). For the notations of ϕ1, ϕ2, ϕ3, LRT, and PTCLT, see [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗
read the original abstract

Due to their parsimony, separable covariance models have been popular in modeling matrix-variate data. However, the inference from such a model may be misleading if the population covariance matrix $\Sigma$ is actually non-separable, motivating the use of statistical tests of separability. The existing separability tests suffer mainly from two issues: 1) test statistics that are not well-defined in high-dimensional settings, 2) low power for small sample sizes and null distributions that depend on unknown parameters, preventing exact error rate control. To address these issues, we propose novel invariant tests using the core covariance matrix, a complementary notion to a separable covariance matrix. We show that testing separability of $\Sigma$ is equivalent to testing sphericity of its core component. With this insight, we construct test statistics that are well-defined in high-dimensional settings and have distributions that are invariant under the null hypothesis of separability, allowing for exact simulation of null distributions. We establish the asymptotic properties of some test statistics by proving the asymptotic spectral equivalence between the sample covariance matrix and its core in a $p/n\rightarrow\gamma\in(0,\infty)$ regime. The large power of our proposed tests relative to existing procedures is demonstrated numerically.

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 proposes a new approach to testing separability of high-dimensional covariance matrices Σ by introducing the complementary notion of a core covariance matrix. It claims that testing separability of Σ is exactly equivalent to testing sphericity of the core component. This equivalence is used to construct invariant test statistics that remain well-defined when p/n → γ ∈ (0, ∞), possess null distributions free of unknown parameters (hence exactly simulable), and satisfy asymptotic spectral equivalence between the sample covariance and the core. Numerical experiments are presented to illustrate higher power relative to existing procedures.

Significance. If the claimed exact equivalence and the asymptotic spectral equivalence hold under the stated high-dimensional regime, the work supplies a principled route to parameter-free, simulable null distributions for separability testing. This would address two documented shortcomings of prior tests (ill-defined statistics in high dimensions and parameter-dependent nulls) and could be useful in matrix-variate applications where Kronecker structure is assumed but must be verified.

major comments (2)
  1. [Abstract and §2] Abstract and §2 (definition of core): the central claim that separability of Σ is equivalent to sphericity of the core holds only if the core is constructed so that any separable Σ maps precisely to a multiple of the identity and any non-separable Σ maps to a non-spherical matrix, without implicit normalizations or additional eigenvalue-separation assumptions. The manuscript must exhibit the explicit mapping and verify that no hidden parameters remain.
  2. [§4] §4 (asymptotic spectral equivalence): the transfer of null-distribution properties to the proposed test statistics relies on proving that the sample covariance and its core are asymptotically spectrally equivalent when p/n → γ. The proof must state the precise moment and eigenvalue conditions under which this equivalence is unconditional; otherwise the invariance and exact simulation claims become conditional.
minor comments (2)
  1. [§2] Notation for the core matrix should be introduced with a clear symbol (e.g., Σ_core) and distinguished from the separable factors at first use.
  2. [Numerical experiments] The numerical section would benefit from reporting the exact simulation size used for critical values and the number of Monte Carlo replications.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and indicate how we will revise the paper accordingly.

read point-by-point responses

  1. Referee: [Abstract and §2] Abstract and §2 (definition of core): the central claim that separability of Σ is equivalent to sphericity of the core holds only if the core is constructed so that any separable Σ maps precisely to a multiple of the identity and any non-separable Σ maps to a non-spherical matrix, without implicit normalizations or additional eigenvalue-separation assumptions. The manuscript must exhibit the explicit mapping and verify that no hidden parameters remain.

    Authors: We agree that the equivalence requires explicit verification. In the revised manuscript we will expand Section 2 to include the full construction of the core component and a direct proof that Σ is separable if and only if its core is a scalar multiple of the identity. The argument will rely solely on the given definitions and will confirm the absence of hidden normalizations or eigenvalue-separation assumptions. revision: yes

  2. Referee: [§4] §4 (asymptotic spectral equivalence): the transfer of null-distribution properties to the proposed test statistics relies on proving that the sample covariance and its core are asymptotically spectrally equivalent when p/n → γ. The proof must state the precise moment and eigenvalue conditions under which this equivalence is unconditional; otherwise the invariance and exact simulation claims become conditional.

    Authors: We accept the need for explicit conditions. We will revise the statement of the main theorem in Section 4 to list the precise assumptions—finite fourth moments of the entries and eigenvalues of Σ bounded away from zero and infinity—under which the asymptotic spectral equivalence holds unconditionally. This will make clear that the invariance and exact-simulation properties are valid under these stated conditions. revision: yes

Circularity Check

1 steps flagged

Equivalence of separability testing to core sphericity is by construction via complementary definition of the core

specific steps
  1. self definitional [Abstract]
    "we propose novel invariant tests using the core covariance matrix, a complementary notion to a separable covariance matrix. We show that testing separability of Σ is equivalent to testing sphericity of its core component."

    The core is introduced as the complementary notion to separability; the claimed equivalence therefore holds exactly by how the core is defined (any separable Σ produces a spherical core and vice versa). The 'show that' step is thus a restatement of the definition rather than a derived result from matrix properties or external theorems.

full rationale

The paper introduces the core covariance matrix as a 'complementary notion to a separable covariance matrix' and then states that testing separability of Σ is equivalent to testing sphericity of the core. This equivalence is load-bearing for the subsequent construction of invariant test statistics and the claim of exact null distribution simulation. Because the core is explicitly positioned as complementary, the mapping (separable Σ maps to spherical core, non-separable maps to non-spherical) is built into the definition rather than independently derived from first principles or external properties. The asymptotic spectral equivalence result between sample covariance and core is presented as a separate technical contribution, but the foundational population-level equivalence reduces to the definitional choice. This produces partial circularity: the central insight is self-definitional while later asymptotic and simulation steps retain independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the definition of the core covariance matrix as a complementary notion to separability and on standard assumptions for covariance matrices in high-dimensional settings. No free parameters are mentioned. The core matrix itself functions as an invented entity whose independent evidence is not provided in the abstract.

axioms (1)
  • domain assumption Covariance matrices are symmetric positive semi-definite and the observations follow a distribution permitting the stated spectral equivalences.
    Standard background assumption invoked for the asymptotic results in the p/n → γ regime.
invented entities (1)
  • core covariance matrix no independent evidence
    purpose: Complementary notion that reduces separability testing to sphericity testing and enables invariant statistics.
    Introduced in the paper as the key device for constructing the new tests.

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    If (j, k) = (u, v), j ̸= k, E[(z⊤ 1jz1k)(z⊤ 1uz1v)] = E[(a⊤b)2] = E   p1X i1,i2=1 ai1bi1ai2bi2   = p1E[a2 1b2 1] + p1(p1 − 1)E[a1b1a2b2] = p1. 38

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  62. [62]

    For instance, if j = u and (j − k)(j − v)(k − v) ̸= 0, E[(z⊤ 1jz1k)(z⊤ 1uz1v)] = E[(a⊤b)(a⊤c)] = E   p1X i1,i2=1 ai1bi1ai2ci2   = nX i1,i2=1 E [ai1ai2] E[bi1]E[ci2] = 0

    Otherwise, using the independence of entries of a, b, c, d yields that E[(z⊤ 1jz1k)(z⊤ 1uz1v)] = 0. For instance, if j = u and (j − k)(j − v)(k − v) ̸= 0, E[(z⊤ 1jz1k)(z⊤ 1uz1v)] = E[(a⊤b)(a⊤c)] = E   p1X i1,i2=1 ai1bi1ai2ci2   = nX i1,i2=1 E [ai1ai2] E[bi1]E[ci2] = 0. One can compute the expectation for other remaining cases using the independence. A...

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  63. [63]

    If (j, k) = (u, v), j = k, E[(z⊤ 1jz2k)(z⊤ 1uz2v)] = E[(a⊤b)2] = p1

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  64. [64]

    If j = k, u = v, and j ̸= u, E[(z⊤ 1jz2k)(z⊤ 1uz2v)] = E[(a⊤b)(c⊤d)] = E[a⊤b]E[c⊤d] = 0

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  65. [65]

    If (j, k) = (u, v), j ̸= k, E[(z⊤ 1jz2k)(z⊤ 1uz2v)] = E[(a⊤b)2] = p1

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  66. [66]

    To prove the second assertion, partition Ip as Ip = diag(Ip1,

    Otherwise, as an analogy to the item 4), one can verify that E[(z⊤ 1jz2k)(z⊤ 1uz2v)] = 0. To prove the second assertion, partition Ip as Ip = diag(Ip1, . . . , Ip1 | {z } p2 ). Then R(Ip) =   W1 ... Wp2   ⇒ R(Ip)R(Ip)⊤ =   W1W ⊤ 1 · · · W1W ⊤ p2 ... ... ... Wp2W ⊤ 1 · · · Wp2W ⊤ p2   , where the row of Wi ∈ Rp2×p2 1 is vec(Ip1) on ith row and ...

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  67. [67]

    To bound W11,11, (I) = nVar (a⊤a)2 = nVar   p1X i=1 a4 i + 2 X 1≤i<j≤p1 a2 i a2 j   ≤ 2nVar p1X i=1 a4 i ! + 16nVar   X 1≤i<j≤p1 a2 i a2 j   ≲ np1ν8 + 16nE[ X i1<j1,i2<j2 a2 i1a2 j1a2 i2a2 j2] ≲ np1ν8 + np4 1 = O(np4 1). (34) 41 Since j = k = u = v = 1, we have that (II) + (III) = 2n(n − 1)Var (a⊤b)2 = 2n(n − 1)Var   p1X i=1 a2 i b2 i + 2 X 1≤i<...

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  68. [68]

    To bound W11,22, (I) = nVar (a⊤a)(b⊤b) ≤ nE[(a⊤a)2(b⊤b)2] = nE[(a⊤a)2]E[(b⊤b)2] = O(np4 1), (II) + (III) = 2n(n − 1)Var (a⊤b)(c⊤d) ≤ n(n − 1)E[(a⊤b)2(c⊤d)2] = 2n(n − 1)E2[(a⊤b)2] ≤ n2 E[ p1X i1=1 a2 i1b2 i1] !2 = O(n2p2 1). (37) 42 On the other hand, (IV) = 4n(n − 1)Cov (a⊤a)(b⊤b), (a⊤c)(b⊤d) = 4n(n − 1) E[(a⊤a)(b⊤b)(a⊤c)(b⊤d)] − E[(a⊤a)(b⊤b)]E[(a⊤c)]E[(b...

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  69. [69]

    (39) As an analogy to (38), one may see that (III) = (IV) = (V) = 0

    To bound W11,12, (I) = nVar (a⊤a)(a⊤b) ≤ nE[(a⊤a)2(a⊤b)2] = nE[ p1X i1,i2,i3,i4=1 a2 i1a2 i2ai3bi3ai4bi4] = nE[ p1X i1,i2,i3=1 a2 i1a2 i2a2 i3b2 i3] = O(np3 1), (II) = n(n − 1)Var (a⊤b)(a⊤c) ≤ n(n − 1)E[(a⊤b)2(a⊤c)2] ≤ n2E[ p1X i1,i2,i3,i4=1 ai1bi1ai2bi2ai3ci3ai4ci4] = n2E[ p1X i1,i3=1 a2 i1b2 i1a2 i3c2 i3] = O(n2p2 1). (39) As an analogy to (38), one may...

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  70. [70]

    Again one may observe that (III) = (IV) = (V) = 0

    To bound W11,23, (I) = nVar (a⊤a)(b⊤c) = nE[(a⊤a)2(b⊤c)2] = nE[(a⊤a)2]E[(b⊤c)2] = O(np3 1), (II) = n(n − 1)Var (a⊤b)(c⊤d) = O(n2p2 1), (40) where the computation of (II) follows from (37). Again one may observe that (III) = (IV) = (V) = 0. Hence, from (40), W11,23 = O(np3 1 + n2p2 1). 43

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  71. [71]

    (41) Also, (36) implies that (III) = n(n − 1)Cov (a⊤b)2, (c⊤d)2 = 0, (IV) = 4n(n − 1)Cov (a⊤b)2, (a⊤c)2 = O(n2p2 1), (V) = 4n(n − 1)(n − 2)Cov (a⊤b)2, (a⊤c)2 = O(n3p2 1)

    To bound W12,12, deduce from (35) and (39) that (I) = nVar (a⊤b)2 = O(np2 1), (II) = n(n − 1)Var (a⊤b)(a⊤c) = O(n2p2 1). (41) Also, (36) implies that (III) = n(n − 1)Cov (a⊤b)2, (c⊤d)2 = 0, (IV) = 4n(n − 1)Cov (a⊤b)2, (a⊤c)2 = O(n2p2 1), (V) = 4n(n − 1)(n − 2)Cov (a⊤b)2, (a⊤c)2 = O(n3p2 1). (42) From (41) and (42), we have that W12,12 = O(n3p2 1)

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  72. [72]

    (43) As the same with the items 3) and 4), one can verify that (III) = (IV) = (V) = 0

    To bound W12,13, (I) = nVar (a⊤b)(a⊤c) = O(np2 1), (II) = n(n − 1)Var (a⊤b)(a⊤c) = O(n2p2 1). (43) As the same with the items 3) and 4), one can verify that (III) = (IV) = (V) = 0. Hence, from (43), W12,13 = O(n2p2 1)

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  73. [73]

    (44) Again we have that (III) = (IV) = (V) = 0

    To bound W12,34, one can deduce from (37) that (I) = nVar (a⊤b)(c⊤d) = O(np2 1), (II) = n(n − 1)Var (a⊤b)(c⊤d) = O(n2p2 1). (44) Again we have that (III) = (IV) = (V) = 0. Thus, (44) implies that W12,34 = O(n2p2 1). Therefore, we verified (30) from the items 1)–7). Using the results from Lemma 4–5, we prove Theorem 5 and Corollary 3. Proof of Theorem 5 No...

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  74. [74]

    47 A.5 Singular Values of R(C) In this subsection, we provide some results on the singular values of R(C)

    log 12− c n3/2δ K2 ! ≲ 2 exp n log 12 − c n3/2δ K2 ! ≲ 2 exp −˜cn3/2δ , where the first ≲ holds because (A1) implies that p2 1 + p2 2 ≍ n, and the second ≲ holds for any constant ˜c ∈ (0, c/K2). 47 A.5 Singular Values of R(C) In this subsection, we provide some results on the singular values of R(C). As discussed in Section 4.3, we show that σ1(R(C)) scal...

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  75. [75]

    We examine whether the above result holds when Σ is a core covariance matrix with a rank −r partial isotropic structure, focusing on r = 1 , 2

    derived the sample eigenvalue bias, as follows (see Theorem 1.1–1.2 of [6]): for j ∈ J i, ℓj a.s.→ ( σ2 ai + γai ai−σ2 , if ai > 1 + √γ, σ2(1 + √γ)2, o.t. . We examine whether the above result holds when Σ is a core covariance matrix with a rank −r partial isotropic structure, focusing on r = 1 , 2. To align with the assumption of Σ, we set λ = 1/(1 + rc/...

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  76. [76]

    Here ( p1, p2, n) = (20, 20, 1600)

    in Proposition 3. Here ( p1, p2, n) = (20, 20, 1600). We conclude this section with the figures that numerically verify the results of Corollary 2 and Theorem 5, as well as the table that demonstrates the consistency of T3. To verify Corollary 2, we generate the test statistic T1 with (p1, p2, n) = (72, 72, 1296) with 1000 simulations, and assess whether ...

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