Optimal Transfer Learning for Missing Not-at-Random Matrix Completion

We thank the reviewer for their valuable feedback, and for highlighting that our claims are supported by rigorous theoretical investigations and numerical experiments. We will fix typos in the revision, and we address all other comments below.

Our methods perform best under new evaluation metrics (MAE and RMSE).

As the Reviewer suggests, we perform new experiments to compare our active and passive sampling methods on Mean Absolute Error (MAE), Root Mean Squared Error (RMSE), as well as the the previously presented metrics of Max Squared Error, and Mean Squared Error (MSE). For ease of comparison, we report the results of Fig. 2 (gene expression transfer) and Fig. 3 (metabolic transfer) again in the tables below. The MAE/RMSE numbers are new.

For the gene expression transfer problem (Figure 2), our methods continue to out-perform baselines with $p_{\textup{Row}} = p_{\textup{Col}} = 0.5$. For gene expression data, the errors are:

Next, we perform the same experiment for the metabolic transfer problem (Figure 3).

We will include the suggested error metrics (MAE and RMSE) in the revision.

Our theoretical upper bounds on Max Squared Error also imply upper bounds on Frobenius error, RMSE, and MAE.

The max-squared error bounds (Theorem 2.6 and Theorem 2.9) immediately imply bounds on Frobenius error, because $\frac{1}{mn} \Vert \hat Q - Q \Vert_{F}^2 \leq \Vert \hat Q - Q \Vert_{max}^2$.

Similarly, both the RMSE and the MAE are upper-bounded by the Max Absolute Error, which can be bounded via Theorems 2.6 and 2.9.

We will include this discussion in the revision.

Our experiments are conducted on datasets where the primary interest is the recovery of entries.

Our real-world experiments (Section 3.1) on gene expression (Fig. 2) and metabolic (Fig. 3) data are on datasets where the primary interest is recovery of missing entries (Parnell et al. 2013, King et al. 2016, Jalan et al. 2024).

We mask matrix entries to compare estimated vs. ground-truth values.

In each experiment, we mask entries of $Q$ for which the ground-truth is known, so that we can compare the estimated $\hat Q_{ij}$ to the ground truth value $Q_{ij}$. We mask according to the active and passive sampling frameworks (lines 302-320, left).

There are many entries of $P, Q$ for which the ground truth is unknown. Without knowing the ground truth, we cannot report estimation error. Therefore we do not perform estimation experiments on missing entries for which ground-truth values are not known.

We thank the reviewer for their valuable feedback, and for highlighting the relevance of our problem (MNAR matrix completion with transfer learning) to biological applications. We address their comments below.

Our minimax results guide algorithm design.

The practical significance of our minimax lower bounds for MNAR matrix completion is that no method can achieve a better estimation error than ours. So, the error guarantees in Theorems 2.6 and 2.9 are unimprovable in our setting.

Exact selection of rows and columns (in active sampling) is realistic for the biological settings we study.

The choice of exact rows and columns to query matches experimental design constraints in multiple settings (lines 9-16, right). Our model is designed to capture these settings. These include (i) metabolite balancing experiments (Christensen and Nielsen, 2000), (ii) gene expression microarrays (Hu et al. 2021), (iii) marker selection for single-cell RNA sequencing (Vargo and Gilbert, 2020) and (iv) patient selection for companion diagnostics (Huber et al. 2022). In Section 3.1, we study precisely the first two settings listed in the introduction: metabolic (Fig. 3) and gene expression (Fig. 2) data.

Some settings, such as electronic health records (Zhou et al. 2023), have different experimental design constraints (lines 46-49, right). Our model would not apply to these settings.

In the revision, we will include additional details on experimental protocols for metabolite balancing and gene expression microarray experiments, as suggested.

Our MNAR model captures the missingness structure of the biological datasets we study.

Our passive sampling model (lines 97-109, left) matches the row/column missingness structure present in gene expression data (Fig. 1, and lines 16-24, right) and metabolic network data (lines 330-342, left). We focus on these settings in our real-world experiments (Section 3.1).

For other kinds of data, other missigness structures may be present (lines 46-49, right). Our methods can be applied to any missingness structure, but we analyze the specific MNAR settings of the paper due to their biological importance.

If the Reviewer has particular missingness structures in mind, we are happy to discuss the applicability of our methods to those settings.

The distribution shift model is realistic for the datasets we study.

The matrix transfer model (Definition 1.2) is commonly used in the biological literature, such as in Genome-Wide Association Studies (see McGrath et al. 2024 and references therein). Our real-world experiments (Section 3.1) further validate that it is appropriate for gene expression (Fig. 2) and metabolic (Fig. 3) transfer problems.

No model applies perfectly to real-world data. It would be interesting to study other models of distribution shift, as we have stated (lines 437-439, right).

Our methods can be applied without any structured transfer assumptions.

Even if Definition 1.2 does not hold, our method can be applied. However, we only analyze its theoretical properties under Definition 1.2.

Our methods can use noisy SVD estimates.

We do not assume clean SVD features for either the source matrix $P$ or target matrix $Q$. The source matrix, which follows Assumption 2.5, can be noisily observed in an MCAR or MNAR fashion (lines 167-172, right). The target matrix is observed with additive noise and has missing rows and columns for both active and passive sampling (lines 79-109, left).

Our active sampler does not assume idealized G-optimal design.

We compute an $\epsilon$-approximate G-optimal design (Definition 2.3) with convex optimization. Specifically, we use the Franke-Wolfe algorithm which runs in polynomial time (Lattimore and Szepesvari 2020). Setting $\epsilon = 0.05$ is sufficient (Theorem 2.6). A perfect $G$-optimal design is not needed.

We will add graph-based and VAE-based references in the revision.

We will discuss VAE-based methods and graph-based methods in the revision, including but not limited to the (see discussion with Reviewer XH4Z) of Ipsen et al. 2020, as well as [1-2] for VAE, and Jalan et al. (NeurIPS 2024) and [3-4] for graphs.

[1] Ghalebikesabi, Sahra, et al. "Deep generative missingness pattern-set mixture models." International conference on artificial intelligence and statistics. PMLR, 2021.

[2] Cai, Hongmin, et al. "Realize generative yet complete latent representation for incomplete multi-view learning." IEEE Transactions on Pattern Analysis and Machine Intelligence46.5 (2023): 3637-3652.

[3] Wang, Yao, et al. "Matrix Completion with Graph Information: A Provable Nonconvex Optimization Approach." arXiv preprint arXiv:2502.08536 (2025).

[4] Zhan, Tong, et al. "Collective Matrix Completion via Graph Extraction." IEEE Signal Processing Letters (2024).

All algorithms are polynomial-time.

We use well-studied polynomial-time algorithms (SVD, least-squares, and Franke-Wolfe).

We thank the reviewer for their valuable feedback, and for highlighting that our paper has strong empirical results, and is comprehensive in its analysis of an interesting problem. We address their comments below.

We will reorganize the writing to first introduce the estimation framework, and then separately discuss the active and passive sampling settings.

Our estimation framework (Section 2.2) involves first learning features from the source matrix via SVD, and then estimating the target matrix via least-squares. This estimation framework applies to both the active sampling (lines 85-96, left) and passive sampling (lines 97-109, left) settings. To improve readability, we will reorganize the problem setup in the revision to first introduce the estimation framework, and then discuss both the active and passive sampling settings. We thank the reviewer for this valuable suggestion.

We thank the reviewer for their valuable feedback, and for highlighting that our paper is well written and presents a sound theoretical framework. We address their comments below.

The manuscript contains results for a non-transfer MNAR baseline (BC22).

The method of BC22 (IEEE Transactions on Information Theory, 2022) is an MNAR imputation method (lines 302-305, left). The manuscript includes this method for all experiments.

Our methods outperform not-MIWAE on MAE, RMSE, MSE, and Max Error.

As the Reviewer suggests, we perform new experiments to compare our methods against the not-MIWAE method of Ipsen et al. 2020. Below, we present the comparison of our active and passive sampling methods on Max Squared Error, Mean Squared Error (MSE), Mean Absolute Error (MAE), and Root Mean Squared Error (RMSE). For ease of comparison, we report the results of Fig. 2 (gene expression transfer) and Fig. 3 (metabolic transfer) again in the tables below. The MAE/RMSE numbers, and the results of not-MIWAE, are new.

For the gene expression transfer problem (Figure 2), our methods out-perform not-MIWAE with $p_{\textup{Row}} = p_{\textup{Col}} = 0.5$. We train not-MIWAE until convergence, with the latent dimension equal to the true matrix rank of $Q$, and a batch size of 32. For gene expression data, the errors are:

Next, we perform the same experiment for the metabolic transfer problem (Figure 3).

As Reviewer XH4Z suggests, our methods may perform better because not-MIWAE is a non-transfer baseline. This further emphasizes the significance of the transfer setting, which our methods capture, as well as the method of LLL22.

We will include both the suggested baseline (not-MIWAE), and the suggested error metrics (MAE and RMSE) in the revision.

Our methods out-perform others under new evaluation metrics (MAE and RMSE).

See the above table. We will include these new metrics in the revision to give a more comprehensive assessment of model performance.

As the reviewer correctly notes, our main focus is on Max Squared Error, which is a commonly used evaluation metric due to its sensitivity to extreme deviations; this is biologically significant.