Matrix Factorization

The core mechanism involves projecting both drugs and diseases into a shared k-dimensional latent space where each dimension represents an abstract biological feature. The rank k is a critical hyperparameter: too low fails to capture complex polypharmacology, while too high risks overfitting to noise in the sparse association matrix. Typical values range from 50 to 200 dimensions for drug repurposing tasks. The dot product between a drug's latent vector and a disease's latent vector yields the predicted association score.

To prevent overfitting on sparse drug-disease matrices, matrix factorization employs L2 regularization (ridge) on the latent factor matrices. The regularization coefficient λ controls the trade-off between fitting observed associations and maintaining small latent vector magnitudes. Advanced variants use graph-regularized matrix factorization, which adds a penalty term encouraging drugs with similar chemical structures or diseases with shared gene signatures to have proximate latent representations, incorporating domain knowledge directly into the optimization objective.

The objective function is typically minimized using Alternating Least Squares (ALS) rather than stochastic gradient descent. ALS exploits the fact that fixing one factor matrix renders the problem convex in the other:

• Fix the disease factor matrix and solve for the drug factor matrix analytically
• Fix the drug factor matrix and solve for the disease factor matrix analytically
• Iterate until convergence This approach handles missing data implicitly by only computing gradients over observed drug-disease pairs, making it computationally efficient for large-scale biomedical matrices.

Standard matrix factorization suffers from the cold-start problem—it cannot predict associations for drugs or diseases not present during training. Inductive Matrix Completion (IMC) addresses this by incorporating side information matrices:

• Drug features: molecular fingerprints, chemical descriptors, or SMILES embeddings
• Disease features: gene expression signatures, phenotype ontologies, or protein interaction profiles IMC learns projection matrices that map these auxiliary features into the shared latent space, enabling zero-shot predictions for novel compounds or newly characterized diseases.

Bayesian Matrix Factorization places prior distributions over the latent factor matrices and likelihood functions over observed associations, enabling uncertainty quantification for each predicted drug-disease pair. This is critical in drug repurposing where false positives carry high costs. The model outputs a predictive distribution rather than a point estimate, allowing researchers to prioritize candidates with both high predicted efficacy and low predictive variance. Markov Chain Monte Carlo (MCMC) or variational inference methods are used for posterior approximation.

Non-Negative Matrix Factorization (NMF) constrains all latent factors to be non-negative, producing an inherently interpretable decomposition:

• Each latent dimension corresponds to an additive, parts-based representation
• Drugs are expressed as positive combinations of latent therapeutic mechanisms
• Diseases are expressed as positive combinations of latent pathological processes This constraint aligns naturally with biological reality where drug effects and disease etiologies are additive rather than subtractive, enabling direct inspection of which latent factors drive a specific repurposing prediction.

An advanced variant of matrix factorization that incorporates side information—such as drug chemical structures and disease gene signatures—into the factorization process. Unlike standard matrix factorization, which cannot handle new entities, inductive matrix completion can predict associations for drugs or diseases not present in the original training matrix by leveraging their auxiliary features. This is critical for real-world drug repurposing where novel compounds or rare diseases must be evaluated.

An unobserved, learned numerical representation that captures hidden properties of drugs or diseases within a low-dimensional vector space. In drug repurposing, latent factors often correspond to implicit pharmacological mechanisms—such as a drug's effect on a specific signaling pathway or a disease's underlying molecular subtype. The dimensionality of the latent space is a critical hyperparameter: too few factors lose signal, while too many introduce noise and overfitting.

The mathematical backbone of many matrix factorization approaches. SVD decomposes the original drug-disease association matrix into three constituent matrices: U (drug latent factors), Σ (singular values), and Vᵀ (disease latent factors). By truncating to the top-k singular values, SVD produces a low-rank approximation that denoises the association matrix and reveals the dominant structure. This denoising effect is particularly valuable for correcting errors in literature-derived drug-disease databases.

An optimization algorithm commonly used to solve matrix factorization problems at scale. ALS works by iteratively fixing one factor matrix and solving for the other, converting a non-convex problem into a sequence of convex least-squares subproblems. This approach is inherently parallelizable and handles implicit feedback well, making it suitable for large pharmaceutical knowledge graphs where most drug-disease pairs have no observed association.

A technique to prevent overfitting by adding penalty terms to the loss function during matrix factorization. Common approaches include:

• L2 regularization (Ridge): Penalizes large latent factor values, encouraging smooth, distributed representations
• Nuclear norm regularization: Directly encourages a low-rank solution by penalizing the sum of singular values In drug repurposing, proper regularization is essential because the association matrix is typically >99% sparse, creating a high risk of fitting noise rather than true pharmacological signal.

The downstream machine learning task that matrix factorization ultimately serves. Given a heterogeneous biomedical network with drugs, diseases, targets, and pathways as nodes, link prediction estimates the probability of a missing edge between two nodes. Matrix factorization approaches this by computing the dot product of the learned latent vectors for a drug-disease pair. A high dot product score indicates a strong predicted association, flagging that pair as a high-priority repurposing candidate for experimental validation.