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Optimal Weighting Proven for Mixing Gene Networks with Proxy Embeddings

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PLANE jointly estimates latent gene positions from a target network and proxy embeddings on a larger gene set, with provably optimal channel weighting and demonstrated gains in network recovery and imputation.

T0 review reviewed 2026-07-09 challenge →

load-bearing objection Gradient-dynamics analysis of network-embedding trade-off is new and mostly sound; block-dominance condition needs verification the 3 major comments →

arxiv 2607.05774 v1 pith:TI7LRTLU submitted 2026-07-07 stat.ME

AI-Augmented Statistical Network Estimation with Proxy Gene Embeddings

Yan Chen , Weijing Tang , Jin-Hong Du This is my paper
classification stat.ME
keywords latent factor modelnetwork estimationproxy embeddingsnormalized gradient descentgene regulatory networksingle-cell CRISPRharmonic mean boundlow-rank factorization
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

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 addresses a practical asymmetry in modern genomics: gene–gene interaction networks are typically measured on a small set of target genes, while externally trained foundation models produce embedding vectors for a much larger gene universe. The authors propose PLANE (Proxy-Latent Assisted Network Estimation), a method that treats the observed network and the proxy embeddings as two noisy observations of a shared low-dimensional latent geometry. The core question is how to set the weight between these two channels. The authors show that a standard zero-order optimality analysis can bound the total reconstruction error but cannot reveal how the weighting trades off network noise against embedding noise inside the latent factors themselves. To get inside that trade-off, they analyze a blockwise normalized gradient descent scheme and prove a deterministic contraction bound on a curvature-weighted, rotation-aligned error metric. Specializing that bound to the target-gene block yields an explicit, closed-form optimal weight that separates network noise from embedding noise. When the unconstrained optimum is admissible, the resulting error satisfies a harmonic-mean identity: it is no worse than the better of the two single-channel errors. The paper validates the rule in simulations and in a CRISPR perturbation dataset, showing that informative embeddings improve latent recovery, network denoising, and link imputation, while cross-validation automatically downweights uninformative proxies.

Core claim

The central discovery is a closed-form optimal weighting formula for balancing a target network against proxy embeddings when both are noisy views of a shared latent structure. By analyzing blockwise Gram-normalized gradient descent through a curvature-weighted, Procrustes-aligned error metric, the authors prove that the target-block estimation error separates cleanly into a network-noise term and an embedding-noise term. The optimal weight is the ratio that equalizes the relative signal-to-noise contributions of the two channels, and the resulting error bound satisfies a harmonic-mean identity, guaranteeing improvement over either channel alone when the unconstrained optimum is admissible.

What carries the argument

The argument rests on three pieces: (1) a joint low-rank factorization in which a target adjacency matrix and a proxy embedding matrix share a common latent factor, identifiable up to orthogonal rotation under mild rank conditions; (2) a blockwise normalized gradient descent algorithm whose Gram-matrix preconditioning produces a clean recursion in an aligned, curvature-weighted error metric coupling three blocks (target latent positions, embedding loadings, and embedding-only latent positions); and (3) a block-dominance condition under which the embedding-loading error becomes asymptotically negligible relative to the target-block error, allowing the aggregate contraction to specialize to a

Load-bearing premise

The block-dominance condition requires that the estimation error for the embedding-loading matrix is asymptotically negligible relative to the target-block error. The paper shows this holds under an additional relative-conditioning condition on the iterates, but does not establish when that condition is met from the data-generating process alone. If the embedding-loading error is not negligible, the clean target-block error bound and the harmonic-mean optimality guarantee may

What would settle it

If the embedding-loading estimation error does not vanish relative to the target-block error as the number of genes grows, the block-dominance condition fails, and the closed-form optimal weight and its harmonic-mean error guarantee no longer hold.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • When externally trained gene embeddings carry biological signal related to the target network, they can be provably combined with the network to improve latent-position recovery beyond what either source achieves alone, with a data-adaptive weight that does not require manual tuning.
  • The harmonic-mean error bound provides a formal guarantee that adding an informative proxy channel never hurts target-block estimation relative to the network-only baseline, as long as the unconstrained optimum is admissible or the projected weight is used.
  • The framework extends covariate-assisted network estimation to a regime where auxiliary information lives on a strictly larger node set than the observed network, enabling link imputation for genes with no observed network data.
  • In the CRISPR perturbation application, the method recovered biologically interpretable regulatory modules (neurotransmitter transport, immune-cell development, hematopoietic control) that were less apparent from the network-only fit, suggesting the approach can surface structure that a single-channel analysis would miss.

Where Pith is reading between the lines

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

  • The harmonic-mean structure of the optimal error suggests a general principle: whenever two noisy channels observe a shared low-rank signal through different linear maps, the optimal combination error may generically take a harmonic-mean form, and the weighting formula could extend to settings beyond gene networks, such as multi-view data integration or sensor fusion with heterogeneous noise.
  • The block-dominance condition—that the embedding-loading estimation error must be asymptotically negligible relative to the target-block error—is the key gatekeeper for the harmonic-mean guarantee. If this condition fails, the clean separation of network and embedding noise may break down, and the optimal weighting could require a more complex formula that accounts for loading-estimation uncertain
  • The framework could be extended to nonlinear link functions (as the authors note) or to settings with more than two channels, where the harmonic-mean identity might generalize to a multi-channel analog, potentially connecting to multi-view canonical correlation analysis.
  • The leave-one-source-out ablation in the real-data analysis suggests that the practical benefit of proxy embeddings depends on complementarity across embedding sources, raising the question of whether the optimal weighting could be further refined by source-specific weights rather than a single aggregate embedding weight.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 7 minor

Summary. This paper proposes PLANE (Proxy-Latent Assisted Network Estimation), a method that jointly leverages a partially observed target gene–gene network and externally learned proxy gene embeddings available on a larger gene set to recover shared latent structure. The core model posits that both the target adjacency and the proxy embeddings are noisy observations of a common low-rank latent factor, and the method balances the two channels through tuning weights (λ₁, λ₂). The paper provides: (i) an identification result (Theorem 1), (ii) a zero-order minimax-optimal bound on the weighted reconstruction loss (Theorem 2), (iii) a gradient-dynamics analysis of blockwise normalized gradient descent yielding deterministic contraction with an explicit statistical tolerance (Theorem 3), and (iv) a target-block error bound that separates network and embedding noise and yields an optimal weighting rule with a harmonic-mean property (Theorem 4 / Corollary B.2). Simulations and a CRISPRa Perturb-seq analysis demonstrate practical gains over network-only baselines.

Significance. The paper addresses a well-motivated problem at the intersection of statistical network analysis and foundation-model embeddings. The theoretical contributions are substantial: Theorem 2 correctly identifies the minimax rate via reduction to Donoho–Gavish, and the gradient-dynamics analysis (Theorems 3–4) is technically involved, requiring careful handling of rotational alignment, blockwise preconditioning, and cross-block interactions. The harmonic-mean bound E(λ*) ≤ min{E(1), E(0)} is a clean and interpretable result. The partial-oracle analysis (Proposition B.2) provides a useful sharpness check. The simulations are reasonably comprehensive, including misspecification robustness and baseline comparisons. The real-data application is biologically grounded. The main theoretical gap—discussed below—is the block-dominance condition in Theorem 4, which is load-bearing but not verified from the data-generating process.

major comments (3)
  1. §3.3, Theorem 4 (Corollary 4 in the text): The block-dominance condition sup_{0≤t≤T} s_B^(t)/s_UQ^(t) = o(1) as n_P → ∞ is load-bearing for the target-block error bound (3.2) and the harmonic-mean weighting result. In the proof (§B.4), this condition is used to drop the o(η s_B^(t)) cross-term in the U_Q-block recursion, which is what enables the clean separation of ||D||_op and ||E_Q||_op in (3.2). Remark 2 (§B.5) shows that curvature-weighted dominance implies Frobenius-norm dominance under an additional relative-conditioning condition on iterates, but neither condition is derived from the data-generating model (1.1). Under the stated asymptotic regime (n_P grows, n_Q fixed), ||E||²_op ~ σ₂²(n_P + d) grows while ||D||²_op ~ σ₁² n_Q stays fixed, so whether s_B/s_UQ → 0 depends on the relative scaling of σ_min(U) (which grows with n_P) versus the noise growth. The paper does not carryout
  2. §3.3 and §B.5, Remark 2: The relative-conditioning condition sup_t (4λ₁||U_Q^(t)||²_op + 2λ₂||B^(t)||²_op)/(2λ₂ σ²_min(U^(t))) = O(1) is introduced in Remark 2 to convert curvature-weighted dominance to Frobenius-norm dominance. This condition implicitly requires σ_min(U^(t)) to scale comparably to ||U_Q^(t)||²_op, which is a strong requirement when n_P grows. The authors should either verify this from the model or explicitly state it as an additional assumption on the data-generating process, separate from the algorithmic regularity conditions in §B.2. As stated, the reader cannot assess when Theorem 4 applies.
  3. §3.3: The practical PLANE-CV procedure selects λ₂ by cross-validation on held-out target-network entries, while the optimal weighting rule λ* depends on population quantities (||U_Q||_op, ||B||_op, σ_min(U_Q), σ_min(B), ||D||_op, ||E_Q||_op) that are not directly observed. The paper does not discuss how to estimate these quantities or whether PLANE-CV consistently selects weights near λ*. A brief discussion of when cross-validation can be expected to approximate the oracle rule, or at least an acknowledgment of this gap and its difficulty, would strengthen the connection between theory and practice.
minor comments (7)
  1. The result in §3.3 is labeled both 'Corollary 4' and 'Theorem 4' in different places (e.g., the statement says 'Corollary 4' but the text refers to 'Theorem 4'). Please use consistent numbering.
  2. §2, Eq. (2.1): The normalization λ₁ + λ₂ = 1 is introduced after the loss function, but Algorithm 1 and the theoretical results use λ₁, λ₂ without this constraint in some places (e.g., the Gram matrices in §3.1 use raw λ₁, λ₂). Clarify where the normalization applies.
  3. §5: The real-data analysis uses d = 10112 concatenated embedding dimensions and n_P = 1099 genes. The ratio d/n_P is large; it would help to comment on whether Assumption 2 (full rank of B with finite condition number) is plausible in this regime, or whether the theory is intended only as an asymptotic guide.
  4. §B.2, Assumption 3: The initialization condition e_0 ≤ τ is stated in terms of the curvature-weighted error metric, which depends on the iterates themselves. A brief comment on how this can be verified in practice (e.g., via spectral initialization) would help.
  5. §C.1, proof of Lemma C.1: There appears to be a stray line 'U^(t)⊤ ∥²_F.' between the B-block and U_{Q^c}-block analyses that seems to be a formatting artifact.
  6. Table D1: The 'Study 2, extra nodes' row lists n_Q = 30 and |Q^c| = n_R^(1), but the text in §4 refers to |Q^c| values up to 500. Clarify the sample size for this study.
  7. The paper would benefit from a brief discussion of computational complexity per iteration of Algorithm 1, particularly the cost of the blockwise Gram matrix inversions.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for a careful and constructive report. The referee correctly identifies the block-dominance condition in Theorem 4 as the main theoretical gap, and we agree that this condition and the relative-conditioning requirement in Remark 2 need to be more explicitly connected to the data-generating model. We also agree that the gap between the oracle weighting rule and the practical PLANE-CV procedure deserves discussion. We address each point below.

read point-by-point responses

  1. Referee: §3.3, Theorem 4: The block-dominance condition sup_{0≤t≤T} s_B^(t)/s_UQ^(t) = o(1) as n_P → ∞ is load-bearing but not verified from the data-generating process. Under the stated asymptotic regime (n_P grows, n_Q fixed), ||E||²_op ~ σ₂²(n_P + d) grows while ||D||²_op ~ σ₁² n_Q stays fixed, so whether s_B/s_UQ → 0 depends on the relative scaling of σ_min(U) versus noise growth.

    Authors: The referee is correct that the block-dominance condition is not derived from the data-generating model (1.1), and we agree this gap should be addressed. We have carried out the analysis the referee requested. Under the asymptotic regime n_P → ∞ with n_Q fixed, and under Assumption 1 (Gaussian noise), we can verify the condition as follows. The B-block curvature-weighted error s_B^(t) involves the term 2λ₂||Δ_B^(t) U^(t)⊤||²_F, which is driven by the statistical error in estimating B. From the B-block gradient (Lemma C.2(ii)), the statistical contribution to s_B scales as λ₂²||E||²_op||U||²_op / σ²_min(U). Meanwhile, the U_Q-block error s_UQ^(t) scales as λ₁²||U_Q||²_op||D||²_op / (2λ₁σ²_min(U_Q) + λ₂σ²_min(B)). Under the standard factor-model scaling where σ_min(U) grows at rate √n_P (which holds when the latent positions have bounded per-row norm), the ratio s_B/s_UQ scales as λ₂²σ₂²(n_P+d)·n_P / [λ₁²σ₁²n_Q · n_P] = λ₂²σ₂²(n_P+d) / (λ₁²σ₁²n_Q), which grows rather than vanishes. This indicates that the block-dominance condition does NOT hold under the standard factor-model scaling without additional structure. However, the condition does hold under a spiked covariance regime where σ²_min(U) grows at rate n_P while ||U||²_op grows at rate n_P, so that the ratio ||E||²_op||U||²_op/σ⁴_min(U) stays bounded. We will revise §3.3 and §B.5 to (a) explicitly state the block-dominance condition as an assumption on the relative signal-to-noise scaling, (b) provide the verification under the spiked covariance regime, and (c) clearly state that the condition is not automatic under the base model and requires this relative scaling. We will also note that the practical relevance of Theorem 4 is supported by the simulation evidence, where the harmonic-mean weighting behavior is empiri revision: yes

  2. Referee: §3.3 and §B.5, Remark 2: The relative-conditioning condition sup_t (4λ₁||U_Q^(t)||²_op + 2λ₂||B^(t)||²_op)/(2λ₂ σ²_min(U^(t))) = O(1) is introduced to convert curvature-weighted dominance to Frobenius-norm dominance. This condition implicitly requires σ_min(U^(t)) to scale comparably to ||U_Q^(t)||²_op, which is strong when n_P grows. The authors should either verify this from the model or explicitly state it as an additional assumption.

    Authors: We agree with the referee that the relative-conditioning condition in Remark 2 should be stated as an explicit assumption rather than introduced only as an interpretive remark. This condition is indeed strong: it requires σ²_min(U^(t)) to be comparable to ||U_Q^(t)||²_op, which under the n_P → ∞ regime requires the smallest singular value of U to grow sufficiently fast relative to the target-block operator norm. We will revise the manuscript to: (1) elevate this condition to a formal assumption (a new Assumption 6) clearly separated from the algorithmic regularity conditions in §B.2, (2) state explicitly that it is a condition on the data-generating process, not merely on the iterates, and (3) note that it holds under the spiked covariance regime described in our response to the first comment, where σ²_min(U) ~ n_P and ||U_Q||²_op ~ n_Q (fixed). We will also add a remark that this condition is the price of converting from the curvature-weighted error metric (which is the natural quantity for the gradient-dynamics analysis) to the unweighted Frobenius-norm error that is more interpretable for practitioners. The referee is correct that without this clarification, the reader cannot assess when Theorem 4 applies. revision: yes

  3. Referee: §3.3: The practical PLANE-CV procedure selects λ₂ by cross-validation on held-out target-network entries, while the optimal weighting rule λ* depends on population quantities that are not directly observed. The paper does not discuss how to estimate these quantities or whether PLANE-CV consistently selects weights near λ*.

    Authors: The referee raises a valid and important point about the gap between the oracle weighting rule λ* (which depends on unobserved population quantities such as ||U_Q||_op, ||B||_op, σ_min(U_Q), σ_min(B), ||D||_op, ||E_Q||_op) and the practical PLANE-CV procedure. We acknowledge that the current manuscript does not establish a formal consistency result connecting PLANE-CV to the oracle rule λ*. This is a genuine theoretical gap that we cannot fully close in this revision. The difficulty is that the oracle rule depends on operator norms of population quantities that are themselves estimated through the iterative procedure, creating a circularity that makes direct plug-in estimation nontrivial. We will add a new paragraph in §3.3 (or §6) that: (1) explicitly acknowledges this gap, (2) explains why a formal consistency result for PLANE-CV → λ* is difficult—because the oracle quantities are functions of the latent factors being estimated, (3) notes that the simulation evidence (Figure 2A–C, D–F) provides empirical support that cross-validation tracks the oracle direction, and (4) suggests this as a direction for future work, potentially via a two-stage procedure where pilot estimates of the population quantities are used to approximate λ*. We believe this honest acknowledgment strengthens rather than weakens the paper. revision: partial

Circularity Check

1 steps flagged

No significant circularity found; one minor self-citation concern that is not load-bearing for the central result.

specific steps
  1. self citation load bearing [Section 3.3, Theorem C.11 (Appendix C.1)]
    "In particular, by Theorem C.11, the error function E(λ) attains its unique global minimum at λ* = ||B||²_op ||E_Q||²_op σ²_min(U_Q) / (||U_Q||²_op ||D||²_op σ²_min(B) + ||B||²_op ||E_Q||²_op σ²_min(U_Q))."

    The optimal weighting rule λ* is derived by applying Theorem C.11, which is a self-contained calculus lemma (minimizing a ratio of quadratic forms over [0,1]) proven within the paper. The formula for λ* is a direct substitution into this lemma's closed-form solution. This is not circular: the lemma is a generic mathematical result, and its application to E(λ) yields the weighting rule as a genuine derivation, not a definition. The concern is that the practical PLANE-CV procedure uses cross-validation rather than this formula, so the theoretical λ* serves as an oracle benchmark. This is a minor self-referential structure but does not constitute circularity.

full rationale

The paper's central derivation chain is largely self-contained and non-circular. Theorem 2 (zero-order bound) is derived from the optimality condition of the weighted loss and a standard trace inequality, then matched to a minimax lower bound from Donoho and Gavish (2014)—an external citation. Theorem 3 (iterative contraction) is derived from a blockwise normalized gradient analysis with explicit algebraic decomposition of directional derivatives (Lemma C.1) and norm bounds (Lemmas C.2–C.5), all proven within the paper from first principles. Theorem 4 (target-block error) follows from Theorem 3 under the block-dominance condition, which is an explicitly stated assumption, not a derived result. The optimal weighting rule λ* is obtained by applying Theorem C.11, a self-contained calculus lemma. The partial-oracle sharpness result (Theorem B.2) is derived from a closed-form least-squares normal equation and Gaussian moment calculations. No step reduces to its inputs by construction. The only minor concern is that the theoretical λ* depends on population quantities not directly observed, but the paper explicitly states that PLANE-CV uses cross-validation independently of this formula, so the practical method is not forced by the theoretical expression. The block-dominance condition (sup s_B/s_UQ = o(1)) is unverified from the data-generating process alone, but this is a correctness risk (assumption may fail), not a circularity issue. Score 2 reflects the minor self-referential structure of the oracle formula being derived from a self-contained lemma, which is standard mathematical practice and not load-bearing for the practical procedure.

Axiom & Free-Parameter Ledger

4 free parameters · 7 axioms · 0 invented entities

The paper introduces no new physical entities, particles, forces, or dimensions. The model uses standard linear algebra objects (latent position matrices, loading matrices, noise matrices). The 'proxy embedding' concept is a repurposing of existing gene embeddings, not a new entity. All axioms are either standard statistical assumptions (Gaussian noise, rank conditions, SNR) or technical conditions for the local convergence analysis (initialization, well-conditioning, block dominance).

free parameters (4)
  • λ₁ = selected by CV or oracle
    Network channel weight; the paper derives an optimal λ* but in practice selects via cross-validation on held-out target-network entries.
  • λ₂ = 0.6 in real-data application
    Embedding channel weight; selected by held-out MSE over a grid. The theoretical optimal value depends on unknown population quantities.
  • k (latent dimension) = 3 in simulations, 8 in real data
    Chosen by eigengap diagnostics for the real data; fixed in simulations.
  • η (step size) = sufficiently small constant per (C.6)
    Algorithmic step size; the theory requires it to be sufficiently small but does not specify an exact value.
axioms (7)
  • domain assumption E[A_Q|U_Q] = U_Q U_Q^⊤, E[W|U] = U B^⊤ (model (1.1))
    The low-rank latent factor model linking network and embeddings through shared latent positions. Invoked in §2 as the data-generating model.
  • domain assumption D_ij and E_ij are i.i.d. mean-zero Gaussian (Assumption 1)
    Used for the minimax lower bound in Theorem 2; the upper bound only requires operator-norm control. Invoked in §2.
  • domain assumption U_Q, B have full column rank with finite condition numbers (Assumption 2)
    Required for identifiability (Proposition 1) and convergence of the iterative procedure. Invoked in §B.2.
  • ad hoc to paper Initial estimator satisfies e_0 ≤ τ (Assumption 3)
    Requires sufficiently accurate initialization for local convergence. The paper notes this could be relaxed via perturbed gradient descent but does not pursue this.
  • domain assumption Signal dominates noise: τ ≥ C₁ × (statistical error terms) (Assumption 4)
    Standard signal-to-noise ratio condition ensuring stable estimation. Invoked in §B.2.
  • ad hoc to paper Iterates remain well-conditioned throughout optimization (Assumption 5)
    Local regularity condition on iterates; holds if initialization is close and step size is small. Invoked in §B.2.
  • ad hoc to paper sup_{0≤t≤T} s_B^(t)/s_UQ^(t) = o(1) as n_P → ∞ (Theorem 4)
    Block-dominance condition required to specialize the aggregate error bound to the target block. Invoked in §3.3 and §B.4.

reviewed 2026-07-09 · how reviews work

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Cite this review

Pith. "Pith review of AI-Augmented Statistical Network Estimation with Proxy Gene Embeddings." pith.science (2026). https://pith.science/paper/TI7LRTLU

@misc{pith2026260705774,
  author       = {Pith},
  title        = {Pith review of: AI-Augmented Statistical Network Estimation with Proxy Gene Embeddings},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TI7LRTLU}},
  note         = {Machine review of arXiv:2607.05774}
}
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read the original abstract

Gene--gene networks are often observed only on a restricted target set, while modern biomedical foundation models provide proxy gene embeddings over substantially larger gene universes. To leverage externally learned representations to improve latent-structure recovery in partially observed target networks, we propose \emph{Proxy-Latent Assisted Network Estimation} (PLANE), an adaptively weighted joint network--embedding latent variable model. PLANE combines the two sources of information through the common latent positions of the target network and proxy embeddings. Under mild rank conditions, the target network enables the identification of latent positions and loading of all nodes up to an orthogonal rotation. We show that zero-order optimality analyses sharply control the weighted reconstruction loss, but are insufficient to identify the optimal weighting. To understand the network and embedding information trade-off for latent-factor recovery, we analyze blockwise Gram-normalized gradient descent and prove deterministic contraction of aligned, curvature-weighted errors up to an explicit statistical tolerance. We then specialize the weighted statistical error bound to derive the target-block error bound, yielding an optimal, data-adaptive choice of the network embedding weights. Simulations and single-cell perturbation analyses show that informative proxy embeddings improve latent recovery, network reconstruction, and imputation beyond the observed target network.

Figures

Figures reproduced from arXiv: 2607.05774 by Jin-Hong Du, Weijing Tang, Yan Chen.

Figure 1
Figure 1. Figure 1: Overview of PLANE. PLANE models both sources as noisy observations of a shared latent biological geometry U, and jointly balances the network and embedding channels to recover target latent positions, denoise the target network, detect gene modules, and impute links. imputation beyond the observed target subnetwork. This distinguishes our setting from standard covariate￾assisted network models and from tra… view at source ↗
Figure 2
Figure 2. Figure 2: Panels A–C show the mean relative Procrustes error of UbQ across weights λ2, with dotted lines indicating oracle choices in study 1. The endpoints correspond to two baseline methods: network-only model (λ2 = 0) and an embedding-only model (λ2 = 1). In study 2, Panels D–E compare informative and null proxy embeddings as |Q c | increases. Panel F reports the mean cross-validated embedding weight. When Q c = … view at source ↗
Figure 3
Figure 3. Figure 3: Results on Norman CRISPR data analysis. (A) The DE-overlap graph among target genes. (B) The held-out relative error on the Norman target graph. (C) The selected λ2 values across repetitions. (D) The estimated target network by PLANE-CV. (E) The estimated target network by network-only PLANE (λ1 = 1). (F) The module sizes and top hubs of PLANE-CV’s target network. are represented by hubs, including ZBTB1, … view at source ↗
Figure 4
Figure 4. Figure 4: Leave-one-source-out ablation of the real-data proxy embedding. Each column removes one embedding source from the full concatenated proxy embedding. (A) Held-out target-network MSE across the λ2 grid; the dotted and dashed horizontal lines denote the network-only and full-embedding references, respectively. (B) Fitted target￾network heatmaps from the selected PLANE-CV fits, with genes ordered by the final … view at source ↗

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This paper was first reviewed by glm-5.2 on July 9, 2026.