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Coupled Tensor-Tensor Completion Method with Applications in Drug Repurposing
Maryam Bagherian, Albert Hung, Ivo Dinov, Joshua Welch
TL;DR
Biomedical tensors often have missing entries, and existing side-information methods assume matrix-form information. CTTC incorporates tensor-valued auxiliary information through a distance-metric-learning approach; the paper reports advantages over existing approaches on real datasets. The study evaluates CTTC against four tensor-completion methods on two benchmark datasets.
Problem
Biomedical tensors can have many unobserved entries, and existing tensor-completion methods with side information assume that the information is in matrix form.
Source: §1 · pp. 1–2Method
Results
The paper reports advantages over existing approaches on real datasets and evaluates CTTC against four state-of-the-art tensor-completion methods on two benchmark datasets.
Source: §1 · p. 2, §6 · p. 14Takeaways & Limitations
CTTC is presented as applicable to sparse, incomplete data and as learning distance metrics for entities such as drugs, targets, or cell lines.
Source: §6 · p. 14The target tensor’s entries are binary and mode labels are non-numeric strings, leaving no additional numerical vectors beyond entry positions for creating artificial side tensors.
Source: §5 · p. 12
Abstract
from arXiv · showhide
Many biomedical challenges can be posed as tensor completion problems where the observed entries of a multidimensional array (a tensor) are used to impute the missing values. In such settings, incorporating side information about the modes of the tensor, such as gene-gene similarity, can significantly enhance the solutions of the completion problem. Most existing tensor completion methods can only incorporate side information in the form of matrices. In this study, we introduce a novel framework to incorporate side information in the form of tensors. Our new approach, called Coupled Tensor-Tensor Completion (CTTC), leverages the hidden connections among multimodal tensors to improve tensor completion performance. In addition to practical utility, CTTC has theoretical foundations in distance metric learning and group theory. We derive an alternating algorithm to solve the CTTC optimization problem and establish its convergence to a stationary point. Finally, we show that CTTC outperforms state-of-the-art tensor completion methods at predicting drug effects. Results: Compared with other tensor completion methods, including HaLRTC, CTRC, Cell, and NTDDR, CTTC demonstrates superior run-time and RSE tensor completion accuracy on two benchmark datasets, DTD and LINCS.
1 Introduction & Related Literature
Biomedical tensor completion addresses sparse measurements, while existing side-information methods assume matrices. CTTC incorporates tensor-valued side information into tensor completion, including relationships with differing mode counts.
1 Introduction & Related Literature
- Many biomedical tensors are mostly unobserved because measuring effects across drugs, cell types, and genes is expensive and time-consuming.A drug’s effect on a particular gene in a particular cell type is one example represented as a three-mode tensor.Source: §1 · p. 1
- Side information can regularize tensor completion; prior approaches include similarity-matrix metric learning and graph-based regularization.Metric-learning methods use similarity matrices as objective regularizers, while graph-based methods incorporate graph regularization into completion or decomposition.Source: §1 · p. 2
- Other related approaches include coupled matrix/tensor factorization and hybrid or coupled-trace-norm constraints.Coupled matrix and tensor factorization uses CP decomposition with coupled matrix information; hybrid methods can also incorporate cross-mode features as coupled matrices.Source: §1 · p. 2
- CTTC couples tensor-valued side information to tensor completion, and its similarity arrays need not have the same number of modes as the target tensor.The method is proposed for cases with multiple kinds of relationships, such as drug-drug, gene-gene, and cell-cell relationships.Source: §1 · p. 2, Figure 1 · p. 3
- Existing tensor-completion methods with side information assume matrix-form side information, although some applications have multiple relationships per tensor mode.The paper’s example is drug-by-cell-type-by-gene data with several kinds of relationships for each mode.Source: §1 · p. 2
2 Background
The background frames tensor completion as low-rank recovery from observed entries and reviews factorization, matricization, and auxiliary-information approaches. It then introduces distance metric learning and the stability of the tensor Mahalanobis distance to auxiliary-tensor perturbations.
2 Background
- Tensor completion estimates an unknown low-rank tensor that matches the observed entries of an incomplete tensor.The background notes that rank is typically approximated with a norm because computing tensor rank is intractable.Source: §2 · p. 3
- The background reviews Tucker completion, PARAFAC models, and low-rank constraints on mode-wise matricizations.Tucker completion can be optimized with block coordinate descent; the described PARAFAC model has a super-diagonal core.Source: §2 · p. 3
- Coupled tensor completion uses matrix- or tensor-valued auxiliary information associated with each mode of the incomplete tensor.The auxiliary arrays can have different sizes and structures from the target tensor.Source: §2 · p. 4
2.1 Distance Metric Learning
- Distance metric learning uses multilinear transformations to define tensor distances; orthogonal mode matrices recover Euclidean distance.The paper generalizes the Mahalanobis distance using a transformation applied along each tensor mode.Source: §2.1 · p. 4
- Small perturbations in auxiliary tensors produce proportionally small changes in the induced tensor Mahalanobis distance.The distance is locally Lipschitz continuous with respect to auxiliary-tensor perturbations on bounded subsets of tensor space.Source: §2.1 · pp. 4–5
3 CTTC Method
CTTC couples an incomplete biomedical tensor with tensor-valued similarity information and uses distance metric learning to support completion. Its alternating optimization has stationary accumulation points under stated assumptions.
3 CTTC Method
- CTTC uses distance metric learning to learn similarities from data and supports auxiliary tensors of varying sizes, rather than restricting side information to matrices.The method treats similarities as inverse distances and allows auxiliary arrays of different sizes.Source: §3 · p. 6
3.1 CTTC Optimization Problem
- The CTTC objective jointly matches observed entries and regularizes tensor completion with mode-specific similarity tensors and transformations.Its variables include the recovered tensor and three transformation matrices; trace-norm penalties are used as convex approximations to rank penalties.Source: §3.1 · pp. 6–7
- The algorithm alternates over the transformation matrices while holding the other two fixed, using trace-norm penalties to regularize them.A trace-norm penalty is also described for the tensor, using a convex combination of its mode matricization trace norms.Source: §3.1 · p. 7
3.2 Convergence Analysis
- Under properness, closedness, and bounded level sets, the objective has a minimizer; the cited sublinear-rate result does not apply to this constrained problem.The minimizer claim is supported by the bounded-level-set argument and Weierstrass theorem.Source: §3.2 · p. 8
- If Assumption 1 holds, the generated sequence is bounded, and each block subproblem has a unique minimum, every accumulation point is stationary for Problem (9).The convergence statement concerns stationarity, not a convergence-rate guarantee.Source: §3.2 · p. 8
4 Datasets and Similarity Tensors
The study evaluates drug-repurposing applications using DTD and LINCS tensors with coupled similarity information. It describes how those tensors and similarities are constructed and identifies the tensor-completion methods used for comparison.
4.1 Datasets
- Drug repurposing uses tensor completion to predict drug performance in new contexts from a small number of previously observed contexts.The example is predicting cell lines where a drug may prove effective using similarity to previously profiled cell lines.Source: §4.1 · p. 8
- DTD is a 810 × 302 × 542 drug–target–disease tensor, while LINCS is a 6780 × 70 × 1104 drug–cell-line–gene tensor.DTD combines association data from multiple sources; LINCS records gene expression after drug treatments across cell lines.Source: §4.1 · p. 9
4.2 Tensor Similarity Calculations
- Similarity scores are computed as 1/(1+d(x,y)); LINCS similarity sources cover chemical, drug, cell-line, gene, and target relationships.The LINCS description lists 25 layers of chemical similarity and multiple sources for gene and cell-line similarities.Source: §4.2 · p. 9
4.3 Other state-of-the-art methods
- The comparison includes NTD-DR, CTRC, Cell, and HaLRTC, spanning graph-regularized, coupled tensor-ring, application-specific, and trace-norm approaches.The described side-information methods are limited to matrix-form information.Source: §4.3 · p. 10
5 Experimental Results
CTTC outperforms the compared tensor-completion methods across missing rates, with particular strength on sparse tensors. The experiments also examine unseen drug or gene prediction, side-information contributions, computational behavior, and subgroup performance in drug-repurposing settings.
5 Experimental Results
- CTTC outperforms the other approaches across missing rates, especially on sparse tensors, where tensor-valued side information may be beneficial.The authors attribute the sparse-tensor advantage as likely due to better incorporation of tensor-valued side information.Source: §5 · p. 11
5.1 Case Study
- On LINCS, results for drug- or gene-held-out entries were comparable to random-missing results, but unseen cell-line prediction was not meaningful with sparse cell-line side information.The held-out drug or gene results support prediction for unseen drugs or genes in the reported setting.Source: Table 3 · p. 11, §5.1 · p. 12
5.2 Ablation Study
- Without side information, DTD average errors were 0.2233 and 0.4190 at missing rates 0.1 and 0.2; progressively adding mode-specific metric tensors improved performance.Adding drug information updates the distance metric used to calculate drug similarities.Source: §5.2 · p. 12, Table 4 · p. 12
5.3 Time, Complexity, and Sensitivity Analysis
- Tensor size n1n2n3 is the dominant complexity factor, while regularization parameters depend on the similarity tensors and auxiliary-information sparsity and reliability.CTTC is reported to converge in relatively small runtime despite its additional computation; runtime comparisons warrant caution because methods use different languages and iteration strategies.Source: §5 · p. 11, §5.3 · p. 13
5.4 Application of CTTC in Drug Repurposing
- CTTC achieved overall RSE 0.0126; in LINCS, stratification used a fixed missing rate of 0.2 across 12 chemical classes and 520 gene families, with Alkyne lowest at 0.0107.Ester, Alkene, and Carboxylic acid class RSEs ranged from 0.0133 to 0.0136.Source: §5.4 · pp. 13–14
- For DTD with 30% of an entire row held out, overall RSE was 0.0877 and performance varied across subgroups; some subgroup RSE values of 0 reflected no masked entries.Disease-category RSE values ranged from 0 to 0.25.Source: §5.4 · p. 14
6 Conclusion
The conclusion presents CTTC as a tensor-completion method evaluated on two benchmark datasets and highlights its joint missing-value prediction and distance-metric learning. It points to applications beyond the demonstrated biomedical tasks and proposes a Riemannian-manifold formulation for future research.
6 Conclusion
- CTTC was evaluated against four state-of-the-art tensor-completion methods on two benchmark datasets.The conclusion introduces the method and describes the scope of its evaluation.Source: §6 · p. 14
- CTTC predicts missing values while learning distance metrics that define similarities among entities such as drugs, targets, or cell lines.The authors describe this joint capability as supporting more accurate predictions and better understanding of the underlying data structure.Source: §6 · p. 14
- The authors report applications in drug-target interaction and cell-line perturbation prediction, while describing CTTC as applicable to other incomplete multidimensional data tasks with auxiliary information.Examples of possible domains include drug discovery, biomarker identification, disease modeling, and personalized medicine.Source: §6 · pp. 14–15
- A proposed future direction is to optimize CTTC directly on suitable fixed-rank tensor manifolds.The authors identify geodesic convexity, sectional curvature, and their influence on optimization and convergence as questions for that formulation.Source: §6 · p. 15