Bias-Corrected Subspace Intersection: Minimax-Optimal Shared Subspace Estimation in Multi-View Data

1 Introduction

• The target is the rank-r subspace shared by two noisy views, separating shared variation from view-specific variation in multi-view data.The model treats each view’s signal space as a population space estimated from observations.Source: §1 · pp. 1–2

1.1 Bias of plug-in subspace intersection

• Plug-in intersection, including AJIVE, is biased because empirical singular vectors leak direction-dependently outside their population signal spaces.Smaller empirical singular values are associated with greater leakage, making the leakage Gram anisotropic and perturbing the intersection.Source: §1.1 · p. 2

1.2 Bias correction and main results

• BCSI reweights empirical singular directions to make leakage approximately isotropic, subtracts the common leakage level, then estimates the shared subspace from the corrected Gram matrix.The weights are motivated by rectangular spiked-matrix theory because empirical singular vectors depend on the noise.Source: §1.2 · pp. 2–3
• Under γ_min ≳ σ√n and comparable shared and view-specific ranks, BCSI attains a minimax rate whose θ^-1 term is unavoidable.The lower bound matches the rate up to universal constants; the θ^-1 term captures added difficulty when view-specific subspaces approach one another.Source: §1.2 · pp. 3–4

1.3 Related work

• Relative to prior JIVE theory, the results allow unequal view dimensions, signal strengths, and view-specific ranks, impose no condition-number restriction, and identify an additional θ^-1 term.Related shared-subspace models often constrain shared directions to be singular vectors or covariance eigenvectors; this model allows arbitrary mixing of shared and view-specific coordinates.Source: §1.3 · pp. 4–5

2.1 Model and problem parameters

• The model sets each signal space to shared and view-specific orthonormal components with independent Gaussian noise; the shared target is their intersection.The loading matrices have a lower bound on their smallest singular value but may otherwise mix shared and view-specific coordinates arbitrarily; positive θ ensures the view-specific spaces have trivial intersection.Source: §2.1 · pp. 5–6

2.2 The BCSI estimator

• BCSI reweights empirical singular directions so their leakage Gram is approximately a common level, then subtracts that level from the within-view Gram blocks.The weights estimate population singular values from empirical singular values using the spike-location relation; below the spectral edge, the estimate stays at the boundary value.Source: §2.2 · pp. 6–7
• The corrected Gram matrix approximates one retaining only within-signal-space components, whose r-dimensional nullspace corresponds to the shared subspace.BCSI selects the r eigenvalues nearest zero and uses their invariant subspace to reconstruct the shared-subspace projector.Source: §2.2 · p. 7
• For unequal signal strengths, BCSI weights the two views so the stronger view contributes more to the reconstruction.The reconstruction uses the corrected factors and the selected near-zero eigenspace.Source: §2.2 · p. 7

3.1 Upper bound

• For the quadratic term, bias correction improves on AJIVE by a factor n/r, especially when θ is small and that term can dominate estimation error.The BCSI bound also has sharper rank dependence in its first-order terms.Source: §3.1 · p. 8
• BCSI’s finite-sample guarantee requires sufficiently strong signals and rank limits, but no condition-number assumption on either signal matrix.The guarantee depends on the smallest signal scales; the stated signal-strength condition supports estimating each view’s signal space.Source: §3.1 · p. 8

3.2 Lower bound and minimax optimality

• When ranks are comparable, the upper and lower bounds match up to constants; for highly unbalanced view-specific ranks, their rank dependence still differs.The upper bound uses the larger individual rank, while the lower bound uses the smaller one.Source: §3.2 · p. 9
• For balanced ranks r ≍ r1 ≍ r2, Theorems 1 and 2 match up to numerical constants and yield the three-term rate in (8); all three terms—the θ-independent, θ^-1/2, and θ^-1 terms—are intrinsic in the signal-strength regime.Source: §3.2 · p. 9
• Even with one view’s signal space known, nearly aligned view-specific directions make the weaker view distinguish shared from individual directions only at a limited scale.The rotation separating those directions is hard to detect when θ is small.Source: §3.2 · p. 10
• Hiding the reference direction in a space of order n leaves only second-order information about the rotation, yielding the new θ^-1 lower-bound obstruction.The paper contrasts this obstruction with prior lower bounds, which capture ambient and θ^-1/2 difficulties but not this additional term.Source: §3.2 · pp. 9–10

4.1 Synthetic data

• The synthetic setup makes view-specific subspaces increasingly aligned, while unequal leakage from empirical singular directions leaves AJIVE’s bias scale constant.Under the stated scaling, the leading BCSI rate term decreases as n^-1/6.Source: §4.1 · pp. 10–11
• BCSI error decreases with dimension and closely follows n^-1/6, while AJIVE and Expected AJIVE errors remain nearly constant.The persistent Expected-AJIVE error indicates systematic bias rather than only random fluctuation; oracle and estimated-parameter BCSI curves are nearly indistinguishable.Source: §4.1 · p. 11, Figure 1 · p. 12

4.2 Real data

• The Nutrimouse comparison supplies AJIVE and BCSI the same marginal and joint ranks, and evaluates genotype and diet associations using paired bootstrap refits.The data comprise gene-expression and liver-fatty-acid views from 40 mice; the labels are used only for evaluation.Source: §4.2 · pp. 11–13
• On Nutrimouse, BCSI has weaker diet association than AJIVE, while genotype association is essentially the same and its paired interval shows no meaningful difference.The paired bootstrap interval favors BCSI for diet.Source: §4.2 · p. 13, Figure 2 · p. 13

5 Analysis of BCSI

• The corrected Gram matrix decomposes into the signal-only oracle Gram matrix plus linear leakage terms and centered quadratic leakage.The weighting makes the leakage Gram close to a common level that is subtracted, bringing the corrected matrix close to the oracle matrix.Source: §5 · p. 14

5.1.1 Oracle geometry

• The oracle Gram matrix has an r-dimensional nullspace encoding coefficient relations that reconstruct the shared subspace, separated from its positive spectrum by a gap.These properties provide the basis for analyzing the perturbed near-zero eigenspace.Source: §5.1.1 · p. 14

5.1.2 Perturbation of the nullspace

• A bound based only on the perturbation’s operator norm is too crude, so the analysis controls the empirical near-zero eigenspace through its residual against the oracle relation.The operator norm of linear leakage terms can scale with the largest signal singular value, introducing an unnecessary upper signal scale.Source: §5.1.2 · p. 15
• Projecting onto the oracle nullspace eliminates the linear leakage terms from the nullspace block, leaving only centered quadratic leakage.The resulting near-zero eigenvectors approximately satisfy the oracle nullspace relation.Source: §5.1.2 · p. 15

5.1.3 From the near-zero eigenspace perturbation to subspace error

• The subspace-error analysis calibrates the near-zero eigenspace to the oracle normalization and separates the error into linear leakage terms and a higher-order remainder.The linear terms depend on leakage, while the remainder contains only second- and higher-order terms.Source: §5.1.3 · p. 16

5.2 Stochastic analysis

• The stochastic analysis controls the leakage terms and the good-event probability; its one-view result gives finite-sample moment control of the weighted leakage Gram under a strong signal condition.The stated condition includes γk ≥ νσ√n; the result justifies subtracting akσ2Ipk in BCSI and supplies inputs for controlling linear and higher-order leakage.Source: §5.2 · pp. 16–17

6 Discussion

• When γmin ≳ σ√n and shared and view-specific ranks are comparable, BCSI attains the minimax rate and distinguishes first- and second-order estimation difficulties.The stated model is two-view JIVE with Gaussian noise.Source: §6 · p. 17
• For substantially unbalanced individual ranks, the upper bound depends on the larger rank while corresponding lower-bound terms depend on the smaller rank.The discussion says the principal-angle decomposition suggests the upper-bound rank dependence may not be sharp.Source: §6 · p. 17
• The current parameter class requires γmin ≳ σ√n, leaving unresolved whether a weaker view can identify shared directions without recovering its entire signal space.The discussion calls for a characterization below the current signal threshold.Source: §6 · p. 18
• The finite-sample analysis relies essentially on Gaussian spiked-matrix behavior and rotational invariance; other noise structures may require estimating and subtracting a non-scalar leakage operator.Source: §6 · p. 18
• For more than two views, the minimax rate remains undetermined as a function of view count, strengths, and individual-subspace alignment.The discussion notes that optimal reconstruction must account for view strengths and the geometry of individual subspaces.Source: §6 · p. 18

Use of AI

• The authors report using GPT 5.6 Sol for editing, formatting, code writing, and proof writing, and Lean 4 to formalize theoretical results.Source: Use of AI · p. 18

A Equivalence of plug-in subspace intersection with AJIVE

• Under the stated eigengap condition, empirical subspace intersection returns the same r-dimensional subspace as two-view AJIVE.Source: A Equivalence of plug-in subspace… · p. 22

B Deterministic ingredients for the upper bound

• The deterministic upper-bound argument combines oracle geometry, relative perturbation, calibration, and an error decomposition into linear leakage and higher-order terms.Source: B Deterministic ingredients for the… · p. 22

B.1.1 Geometry of the joint signal spaces

• The joint signal-space geometry includes eigenvalue-1 directions for the |r1 − r2| unpaired directions when individual ranks differ.Source: §B.1.1 · p. 23

B.1.2 Geometry of the oracle Gram matrix

• The oracle Gram matrix has nullity r because its kernel equals ker(S), with rank(S) determined by the shared intersection of the two signal spaces.Source: §B.1.2 · p. 24

B.1.3 Population norm bounds

• The population norm bounds use the oracle geometry to control pseudoinverse-related quantities and matrix ranks.Source: §B.1.3 · p. 24

B.3.1 Calibration

• Calibration changes the empirical eigenspace basis without changing its column space, aligning it with the oracle normalization.Source: §B.3.1 · p. 27

B.3.3 Deterministic error decomposition

• The shared-subspace error separates into direct and eigenspace-displacement contributions linear in leakage, plus a higher-order remainder.The displacement contribution includes −T LQ⋆; the remainder is bounded through four terms.Source: §B.3.3 · pp. 29–30

C.1 Gaussian concentration and matrix bounds

• Gaussian concentration and matrix-deviation bounds control moments of independent bilinear forms and centered same-sample quadratic forms.The coefficient-moment bounds also hold conditionally when the deterministic matrix is independent of the displayed Gaussian matrices.Source: §C.1 · p. 31

D.1 Normalized model and theorem

• The bias-correcting weights target a diagonal leakage-Gram bias that varies with the signal singular values.The empirical singular-vector leakage depends on the corresponding population singular value.Source: §D.1 · p. 33
• Theorem 3 gives a quantitative weighted-leakage isometry after weighting aims to make the leakage Gram matrix close to the same scalar matrix across signal directions.The normalized leakage Gram is centered at βI_p.Source: §D.1 · pp. 33–34

D.2 Proof of Lemma 4

• The proof of Lemma 4 applies Theorem 3 to each view after normalizing by its noise scale and dimension, then translates the weighted quantities back to original units.The rescaling yields C_k = σ√b_k Φ_kW_L and L_k = σ√b_k U_⊥W_L.Source: §D.2 · p. 34

• For wide matrices, the proof combines localization with a radial estimate to establish the left-leakage bound.It then obtains the radial bound and localization probability from separate lemmas.Source: §D.3 · pp. 35–37
• Approximate linear relations couple the left and right leakage Grams, and inversion yields cancellation identities used to control the left leakage.The proof forms a linear system because the empirical singular-vector matrices have orthonormal columns.Source: §D.3 · p. 36
• A corresponding right-leakage bound for wide matrices supplies the control needed to treat tall matrices.The tall case is reduced to a rescaled transpose using Gaussian rotational invariance.Source: §D.3 · pp. 35, 37
• Under transposition, the right singular vectors of the rescaled matrix are the original left singular vectors, linking its right-leakage control to the tall-case left-leakage bound.The proof identifies the transformed weights and singular values before applying the bound.Source: §D.3 · p. 37

D.4 Proof of Lemma 15

• The localization proof places selected empirical singular values on the upper branch of the weighting map and shows the initial event implies the localization event.The argument uses signal conditions to bound the empirical singular values and their corresponding τ coordinates.Source: §D.4 · p. 38

D.5 Proof of Lemma 16

• On the localization event, the lower singular-vector equations admit a convergent series whose Gram expansion links leakage to the in-signal components.The proof also identifies noise-dependent coefficient terms as a difficulty in controlling these Grams.Source: §D.5 · pp. 38–39

D.5.1 Centering the non-bulk noise

• Conditioning on the bulk noise matrix separates Gaussian coefficient matrices into conditional means and centered fluctuations, which are collected into a remainder.The conditional means form the empirical Gram operators.Source: §D.5.1 · pp. 39–40

D.5.2 Replacing the bulk traces

• The empirical operators’ remaining randomness comes through normalized traces of bulk-noise functions, which the proof replaces by Marchenko–Pastur limits.The difference between empirical and population operators is attributed to this trace replacement.Source: §D.5.2 · pp. 41–42

• Deterministic comparisons among the balancing weights convert the common estimate into the left- and right-leakage bounds of Lemma 16.The proof states the resulting forms in (78) and (79).Source: §D.5.3 · pp. 42–43
• The proof controls both coefficient-centering fluctuations and errors from replacing empirical bulk traces with deterministic limits.These are the two error sources isolated before bounding them.Source: §D.5.3 · pp. 42–43
• Bounds for the two errors are combined into a balanced estimate for ε_U and ε_V.The proof uses ε_U = R_U,coef + R_U,bulk and ε_V = R_V,coef + R_V,bulk.Source: §D.5.3 · p. 43

D.6 Proof of Lemma 17

• The left leakage satisfies ∥U⊥WL∥L128 ≲ √β, whereas the right leakage satisfies ∥V⊥WR∥L128 ≲ 1, as established in the cited analysis.Source: §D.6 · p. 44

D.7.1 Proof of Proposition 1

• Contour-residue calculations establish the entrywise formulas for AU, B, and AV in Proposition 1.For AU and B, the resulting entries are βrirjMij/(τiτj − β) and βMij/(τiτj − β), respectively.Source: §D.7.1 · p. 45

D.7.2 Proof of Lemma 18

• Centered Gaussian quadratic and bilinear coefficients yield geometrically summable bounds, and combining them proves (102).The coefficient matrices have rank at most a; their operator norms are controlled by C(1 + ∥Z22∥op)^4 times the corresponding power of ∥Hbulk∥op.Source: §D.7.2 · p. 46

D.7.3 Proof of Lemma 19

• The cutoff-resolvent argument bounds the fixed-point residual by O(a^-1) and gives a uniform contour error of O(1/√a).The proof uses Gaussian concentration, integration by parts, and stability of the Marchenko–Pastur equation.Source: §D.7.3 · p. 50

D.7.4 Proof of Lemma 20

• The four bulk operator differences are bounded by C√a/b, which yields (104).The base term Σ^-1MΣ^-1 cancels between the empirical and population operators for one of the differences.Source: §D.7.4 · p. 51

E.1 Proof of (34a)

• For the T-multiplied term in (34a), submultiplicativity is too crude, so the proof uses rotational invariance of Lk to obtain rank dependence.The direct bound would retain a √ak factor from Lk.Source: Step 2: Bound T · p. 51

Step 2: Bound T

• Lemma 21 uses invariance under orthogonal transformations of Lk to bound the product using operator and Frobenius norms of the matrices on either side.Its setup takes E and the matrices Ψ and Θ to be Hk-measurable.Source: Step 2: Bound T · p. 51
• For Step 2, Hk-measurability suffices to apply the concentration bound, and norm bounds on T and Rk control the resulting terms.The passage states that Ψ and Θ need not be independent of Lk; their Hk-measurability is sufficient.Source: Step 2: Bound T · pp. 51–52

E.2 Proof of (34b)

• (34b) is proved by bounding ∆quad, Pcol(S)L, and ∆+0, then using pmax = r + (r1∨r2) and the small-rate assumptions.On Eemb, col(S) = M1 + M2; the proof applies concentration to measurable blocks and controls diagonal and off-diagonal terms.Source: §E.2 · pp. 52–53

E.3 Proof of (34c)

• (34c) follows by bounding ∆00 and ∆++, using the prior bound for ∆+0, and applying exponent-32 Markov control under the small-rate assumption.Since SP0 = 0, terms in ∆ containing one factor L vanish when projected by P0 on both sides.Source: §E.3 · pp. 53–54

E.4 Proof of Lemma 21

• Rotational invariance randomizes the orientation of Lk while preserving Hk, reducing the concentration proof to a fixed-matrix Haar estimate.The proof combines that estimate with Gaussian comparison and concentration bounds.Source: §E.4 · pp. 54–55

F Probabilistic tools for the lower bound

• The lower-bound toolkit includes a Hellinger path bound and posterior-mean bounds controlled by mutual information.For the Gaussian mean channel, the posterior-mean bound is sharpened using an information bound proportional to λ²/σ².Source: F Probabilistic tools for the lower… · pp. 57–58

G Proof of Theorem 2

• The first lower-bound term concerns shared-subspace estimation with known view-specific spaces; the other two concern distinguishing shared directions when those spaces are nearly aligned.The latter terms capture added difficulty from near alignment.Source: G Proof of Theorem 2 · p. 59

G.1 Proof of (140a)

• Fano’s argument yields the minimax risk lower bound in (140a) from separated shared-direction alternatives and comparisons of their observation laws.The construction fixes the view-specific subspaces and all but one shared direction.Source: §G.1 · pp. 59–60

G.2 A common construction for the two θ-dependent terms

• A rank-one Assouad family makes neighboring hypotheses differ in one shared direction and one orthogonal direction, while View 1 has the same distribution across hypotheses.Fixing Z gives the θ^-1/2 term, whereas mixing over Haar-distributed Z gives the θ^-1 term.Source: §G.2 · pp. 61–63

G.3 Proof of (140b)

• For the fixed-Z construction, the leading change in View 2’s covariance is proportional to ρ⊥, with δ ≍ σ/(γ2ρ⊥).The proof identifies ρ⊥² ≍ θ in deriving (140b).Source: §G.3 · pp. 63–64

G.4 Proof of (140c)

• Randomizing the unobserved View-2 directions yields the √n factor and, with the chosen separation, establishes the minimax lower bound (140c).Conditioning on q − 1 columns leaves one direction uniform on an M-dimensional sphere; its M−1 information factor produces the √n rate factor.Source: §G.4 · pp. 64, 66

G.5 Proofs of the neighboring-law lemmas

• The neighboring-law proofs reduce comparisons to View 2, using covariance-projector laws or Haar mixtures of Gaussian covariance or mean laws, depending on the construction.The regular and hidden low-loading cases use Gaussian loadings; the high-loading case uses deterministic loadings.Source: §G.5 · p. 66

G.5.1 Proof of Lemma 28

• For Lemma 28, the Gaussian-loading comparison bounds the View-2 total variation distance by 1/8 when δ ≤ cσ/(γ²ρ⊥), yielding the lemma after accounting for conditioning.The proof compares covariance projectors after integrating out the unconditioned Gaussian loading.Source: §G.5.1 · p. 68

G.5.2 Proof of Lemma 29

• The proof reduces the unconditioned laws Q0 and Q1 to an (M + 2)-dimensional covariance mixture and compares them along a Hellinger path to show TV(Q0, Q1) ≤ 1/8.Because ez is not observed, the mixture score cannot be computed as if ez were observed; a lemma controls it.Source: §G.5.2 · pp. 68, 70, 71

G.5.3 Proof of Lemma 30

• For Lemma 30, the hidden high-loading comparison reduces View 2 to an (M + 2) × 2 Gaussian mean model and compares the resulting mixtures along a Hellinger path.This construction uses deterministic loading, so it requires no conditioning-error reduction.Source: §G.5.3 · p. 71
• A rotation-path Hellinger comparison with φ ≤ 4δ yields TV(Q0, Q1) ≤ 1/4 for sufficiently small c, completing Lemma 30.The proof then combines this distance bound with the equality between the joint and View-2 distances.Source: §G.5.3 · pp. 72–73

G.6.1 Proof of Lemma 31

• To prove (186), the argument uses Fisher’s identity, bounds I(ez; W) by C I ρ^2, and controls the needed sample covariance uniformly in t.The proof combines these estimates to obtain the bound in (186).Source: §G.6.1 · pp. 73–75

G.6.2 Proof of Lemma 32

• For Lemma 32, equal rotations cancel ρ-independent score terms; the remaining latent-direction dependence is a Gaussian mean channel with signal strength s2ρ⊥, to which Lemma 26 applies.The proof then conditions on the channel coordinates to take a score second moment.Source: §G.6.2 · pp. 75–76