Deep Matrix Factorization

• Deep Matrix Factorization is a multilayer linear modeling approach that generalizes classical matrix factorization by representing a target matrix as a product of multiple factors for hierarchical representation.
• It enforces constraints like nonnegativity, sparsity, and orthogonality to maintain interpretable latent structures, enhancing clustering and community detection across applications.
• Optimization methods range from alternating updates to end-to-end backpropagation, with implicit regularization guiding convergence to low-rank, interpretable solutions.

Deep Matrix Factorization (DMF) extends classical constrained low-rank matrix approximation to multilayer linear factorizations, with the aim of combining the interpretability of matrix factorizations with hierarchical feature extraction. In its canonical form, a target matrix is represented as a product of multiple factors, so that depth becomes a structural property of the factorization rather than a synonym for nonlinear activations. In adjacent recommendation-system literature, the same expression is also used for models that replace the fixed inner product of matrix factorization by a learned nonlinear function or by deep encoders that generate user and item factors. Across these usages, the central themes are low-rank structure, hierarchical representations, constrained latent spaces, and the implicit or explicit regularization induced by the optimization procedure (Handschutter et al., 2020, Zhang, 2022, Dziugaite et al., 2015).

1. Canonical formulations

Classical matrix factorization starts from a shallow approximation such as $X \approx WH$, or, in recommendation form, $$\hat r_{ui} = \mu + b_u + b_i + \mathbf p_u^\top \mathbf q_i$$. DMF replaces the single factor pair by a chain of factors. A standard end-to-end formulation is

$$\min_{\{W_\ell\},H}\; \frac{1}{2}\|X - W_1W_2\cdots W_L H\|_F^2 + \sum_{\ell=1}^{L} R_{W_\ell}(W_\ell) + R_H(H),$$

while a right-deep variant factors the code or basis recursively rather than only the end-to-end map (Handschutter et al., 2020).

In the deep linear setting, DMF is equivalent to deep linear network training. A canonical objective is

$$\mathcal{L}(W_1,\dots,W_L) := \bigl\| M - W_LW_{L-1}\cdots W_1 \bigr\|_F^2,$$

and the same analysis applies to linear networks after absorbing the data matrix into the first factor (Kamber et al., 30 Sep 2025). This equivalence is important because it connects matrix factorization to results on optimization geometry, sharpness, balancing, and implicit bias.

The nonnegative and semi-nonnegative branches specialize this template by enforcing sign constraints on some or all factors. Deep Semi-NMF uses

$$X^{\pm} \approx Z_1^{\pm} Z_2^{\pm} \cdots Z_m^{\pm} H_m^{+},$$

with nonnegative intermediate representations so that each layer retains a clustering interpretation, while deep NMF further constrains all factors to be elementwise nonnegative (Trigeorgis et al., 2015, Leplat et al., 2023).

A graph-specialized formulation appears in multimodal connectome analysis. Supervised Deep Multimodal Matrix Factorization (SD3MF) represents each subject-level graph by

$A_i^{(m)} \approx \Psi^{(m)} S_i \Psi^{(m)\top},$

where $\Psi^{(m)} = \prod_{\ell=1}^{L} W_\ell^{(m)}$ is a modality-specific deep nonnegative membership map and $S_i$ is a subject-specific community interaction matrix shared across modalities (Seyedi et al., 13 May 2026).

A distinct but related line in recommendation treats DMF as a learned interaction function. One generic form is

$\hat r_{ui} = f_\theta(\mathbf p_u,\mathbf q_i),$

and NNMF instantiates this idea with a multi-layer feed-forward neural network whose inputs are user-specific features, item-specific features, and learned interaction channels, thereby replacing the fixed inner product by a learned nonlinear map (Zhang, 2022, Dziugaite et al., 2015).

2. Structural constraints and model families

The usefulness of DMF depends heavily on constraints. Without them, adjacent factors can absorb arbitrary invertible transformations, and a deep product can collapse into a shallow map. The literature therefore relies on nonnegativity, approximate orthogonality, sparsity, graph Laplacian regularization, simplex constraints, minimum-volume penalties, and symmetry or sign structure, depending on the application (Handschutter et al., 2020).

Deep Semi-NMF retains mixed-sign basis matrices and nonnegative hidden codes, so that each $H_\ell$ can be interpreted as a soft clustering according to a latent attribute. The semi-supervised extension Deep WSF adds layer-wise graph Laplacian terms,

$$\hat r_{ui} = \mu + b_u + b_i + \mathbf p_u^\top \mathbf q_i$$0

which allows partial prior information for different attributes to shape distinct layers of the hierarchy (Trigeorgis et al., 2015).

Deep approximately orthogonal NMF imposes nonnegativity together with approximate orthogonality to encourage cluster-indicator structure. Its deep factorization is written as

$$\hat r_{ui} = \mu + b_u + b_i + \mathbf p_u^\top \mathbf q_i$$1

with an additional penalty on pairwise correlations among columns of the $$\hat r_{ui} = \mu + b_u + b_i + \mathbf p_u^\top \mathbf q_i$$2 factors. This construction is explicitly motivated by hierarchical clustering through different levels of abstraction (Qiu et al., 2017).

Sparse deep NMF introduces column-wise sparsity on selected factors. The proposed SDNMF/L, SDNMF/R, SDNMF/RL1, and SDNMF/RL2 variants place penalties on columns of $$\hat r_{ui} = \mu + b_u + b_i + \mathbf p_u^\top \mathbf q_i$$3, on columns of $$\hat r_{ui} = \mu + b_u + b_i + \mathbf p_u^\top \mathbf q_i$$4, or on both, in order to learn localized features or more discriminative sample representations. The same work also considers nonlinear extensions and reports that the root nonlinearity performed best among tanh, root, sigmoid, and softplus (Guo et al., 2017).

A different axis of variation concerns the loss itself. Deep NMF with $$\hat r_{ui} = \mu + b_u + b_i + \mathbf p_u^\top \mathbf q_i$$5-divergences replaces the least-squares error by

$$\hat r_{ui} = \mu + b_u + b_i + \mathbf p_u^\top \mathbf q_i$$6

with particular attention to the Kullback-Leibler divergence. The same framework also introduces a minimum-volume version with $$\hat r_{ui} = \mu + b_u + b_i + \mathbf p_u^\top \mathbf q_i$$7 regularization, motivated by identifiability and interpretability in domains such as topic modeling and hyperspectral unmixing (Leplat et al., 2023).

In graph completion, DMF can be regularized by spectral geometry. Spectral Geometric Matrix Completion augments a deep linear factorization $$\hat r_{ui} = \mu + b_u + b_i + \mathbf p_u^\top \mathbf q_i$$8 with a Dirichlet energy

$$\hat r_{ui} = \mu + b_u + b_i + \mathbf p_u^\top \mathbf q_i$$9

thereby incorporating row and column graph structure without breaking the DMF form (Boyarski et al., 2019).

SD3MF combines several of these ideas. It enforces nonnegativity on all deep factors, normalizes rows of $$\min_{\{W_\ell\},H}\; \frac{1}{2}\|X - W_1W_2\cdots W_L H\|_F^2 + \sum_{\ell=1}^{L} R_{W_\ell}(W_\ell) + R_H(H),$$0 so that $$\min_{\{W_\ell\},H}\; \frac{1}{2}\|X - W_1W_2\cdots W_L H\|_F^2 + \sum_{\ell=1}^{L} R_{W_\ell}(W_\ell) + R_H(H),$$1, allows signed $$\min_{\{W_\ell\},H}\; \frac{1}{2}\|X - W_1W_2\cdots W_L H\|_F^2 + \sum_{\ell=1}^{L} R_{W_\ell}(W_\ell) + R_H(H),$$2 to capture positive and negative inter-community couplings, learns simplex-constrained modality weights $$\min_{\{W_\ell\},H}\; \frac{1}{2}\|X - W_1W_2\cdots W_L H\|_F^2 + \sum_{\ell=1}^{L} R_{W_\ell}(W_\ell) + R_H(H),$$3, and symmetrizes $$\min_{\{W_\ell\},H}\; \frac{1}{2}\|X - W_1W_2\cdots W_L H\|_F^2 + \sum_{\ell=1}^{L} R_{W_\ell}(W_\ell) + R_H(H),$$4 in undirected settings (Seyedi et al., 13 May 2026).

3. Optimization, implicit regularization, and landscape geometry

Two broad training regimes recur throughout the literature: greedy layer-wise factorization and joint end-to-end optimization. The sequential regime factors one layer at a time, as in multilayer NMF, while end-to-end methods optimize the full product directly and allow deeper layers to retroact on earlier ones (Handschutter et al., 2020). Both remain nonconvex, but the choice changes the geometry seen by the optimizer.

Classical constrained DMF algorithms rely on alternating optimization. Representative updates include multiplicative rules, HALS, projected gradient, proximal gradient, ADMM, and block coordinate descent. Sparse deep NMF uses Nesterov’s accelerated gradient with convergence rate $$\min_{\{W_\ell\},H}\; \frac{1}{2}\|X - W_1W_2\cdots W_L H\|_F^2 + \sum_{\ell=1}^{L} R_{W_\ell}(W_\ell) + R_H(H),$$5 after $$\min_{\{W_\ell\},H}\; \frac{1}{2}\|X - W_1W_2\cdots W_L H\|_F^2 + \sum_{\ell=1}^{L} R_{W_\ell}(W_\ell) + R_H(H),$$6 steps iteration for the convex subproblems arising in block updates (Guo et al., 2017). Deep NMF with $$\min_{\{W_\ell\},H}\; \frac{1}{2}\|X - W_1W_2\cdots W_L H\|_F^2 + \sum_{\ell=1}^{L} R_{W_\ell}(W_\ell) + R_H(H),$$7-divergences uses block majorization-minimization and ADMM inner loops for minimum-volume KL models (Leplat et al., 2023). Multi-view clustering methods such as MVC-DMF-PA and DMFAW use alternating optimization with orthogonal Procrustes steps, multiplicative updates, and closed-form view-weight updates, with monotonic decrease and convergence to a stationary point stated for the block scheme (Zhang et al., 2021, Khalafaoui et al., 2024).

Other models are trained by backpropagation rather than multiplicative updates. SD3MF is explicit on this point: it is trained via backpropagation, no multiplicative NMF-style closed-form updates are used, and optimization proceeds by end-to-end stochastic gradient descent with learning rate $$\min_{\{W_\ell\},H}\; \frac{1}{2}\|X - W_1W_2\cdots W_L H\|_F^2 + \sum_{\ell=1}^{L} R_{W_\ell}(W_\ell) + R_H(H),$$8, up to $$\min_{\{W_\ell\},H}\; \frac{1}{2}\|X - W_1W_2\cdots W_L H\|_F^2 + \sum_{\ell=1}^{L} R_{W_\ell}(W_\ell) + R_H(H),$$9 iterations, together with ReLU and row normalization to enforce nonnegativity and fuzzy memberships (Seyedi et al., 13 May 2026).

A major theoretical theme is implicit regularization. Gradient descent or gradient flow on deep linear parameterizations exhibits a dynamical bias towards low-complexity, low-rank solutions, and this bias strengthens with depth. In matrix sensing and completion, the singular values of the end-to-end product evolve according to a depth-dependent law, so that larger singular values accelerate while smaller ones move slowly; this widens spectral gaps and strengthens the low-rank bias (Arora et al., 2019). This perspective also shows why depth can improve recovery even when the factorization is formally overparameterized.

Recent work makes the geometry near minimizers explicit. For overparameterized DMF with squared-error loss, the maximum eigenvalue of the Hessian at any global minimizer has the exact form

$$\mathcal{L}(W_1,\dots,W_L) := \bigl\| M - W_LW_{L-1}\cdots W_1 \bigr\|_F^2,$$0

where $$\mathcal{L}(W_1,\dots,W_L) := \bigl\| M - W_LW_{L-1}\cdots W_1 \bigr\|_F^2,$$1 and $$\mathcal{L}(W_1,\dots,W_L) := \bigl\| M - W_LW_{L-1}\cdots W_1 \bigr\|_F^2,$$2 are layerwise suffix and prefix products at the minimizer (Kamber et al., 30 Sep 2025). This yields an exact edge-of-stability threshold $$\mathcal{L}(W_1,\dots,W_L) := \bigl\| M - W_LW_{L-1}\cdots W_1 \bigr\|_F^2,$$3 for gradient descent near a minimum.

Under $$\mathcal{L}(W_1,\dots,W_L) := \bigl\| M - W_LW_{L-1}\cdots W_1 \bigr\|_F^2,$$4 regularization, the landscape changes again. Regularized DMF with squared-error loss admits a unique end-to-end minimizer for all target matrices subject to factorization, except for a Lebesgue measure-zero exceptional set formed by the depth and the regularization parameter. In the scalar case, the Hessian spectrum is constant across all minimizers; in the matrix case, the Frobenius norm of each layer is constant across all minimizers, and a critical threshold for the regularization parameter causes the unique end-to-end minimizer to collapse to zero (Kamber et al., 28 Mar 2026).

4. Interpretability, clustering structure, and identifiability

Interpretability is one of the main reasons DMF remains distinct from generic deep representation learning. In nonnegative and semi-nonnegative models, factors retain a direct relation to parts, attributes, prototypes, or communities. Semi-NMF already admits a soft clustering interpretation, and Deep Semi-NMF was introduced precisely to learn hidden representations that allow interpretation according to different, unknown attributes of a dataset (Trigeorgis et al., 2015).

Orthogonality strengthens that interpretation. In approximately orthogonal NMF, the $$\mathcal{L}(W_1,\dots,W_L) := \bigl\| M - W_LW_{L-1}\cdots W_1 \bigr\|_F^2,$$5 factor behaves like a cluster-indicator matrix, and the deep variant propagates this structure across layers to realize hierarchical clustering. This supports the view that deep factorizations can yield progressively coarser abstractions while preserving nonnegativity and cluster separability (Qiu et al., 2017).

In multimodal graphs, SD3MF makes the interpretability mechanism explicit. The matrices $$\mathcal{L}(W_1,\dots,W_L) := \bigl\| M - W_LW_{L-1}\cdots W_1 \bigr\|_F^2,$$6 organize ROIs into communities, $$\mathcal{L}(W_1,\dots,W_L) := \bigl\| M - W_LW_{L-1}\cdots W_1 \bigr\|_F^2,$$7 summarizes intra-community and inter-community coupling for subject $$\mathcal{L}(W_1,\dots,W_L) := \bigl\| M - W_LW_{L-1}\cdots W_1 \bigr\|_F^2,$$8, and the encoder-side quantity

$$\mathcal{L}(W_1,\dots,W_L) := \bigl\| M - W_LW_{L-1}\cdots W_1 \bigr\|_F^2,$$9

provides a directly data-projected community summary. The model further supports hard assignments $$X^{\pm} \approx Z_1^{\pm} Z_2^{\pm} \cdots Z_m^{\pm} H_m^{+},$$0, ROI saliency scores from $$X^{\pm} \approx Z_1^{\pm} Z_2^{\pm} \cdots Z_m^{\pm} H_m^{+},$$1, and group-wise reconstruction with two-sample $$X^{\pm} \approx Z_1^{\pm} Z_2^{\pm} \cdots Z_m^{\pm} H_m^{+},$$2-tests to highlight discriminative connections (Seyedi et al., 13 May 2026).

Identifiability is more subtle. The 2020 survey notes that direct deep identifiability results are scarce and that recursive identifiability often reduces to shallow identifiability at each layer (Handschutter et al., 2020). SD3MF gives a concrete separability result: if $$X^{\pm} \approx Z_1^{\pm} Z_2^{\pm} \cdots Z_m^{\pm} H_m^{+},$$3, $$X^{\pm} \approx Z_1^{\pm} Z_2^{\pm} \cdots Z_m^{\pm} H_m^{+},$$4, and $$X^{\pm} \approx Z_1^{\pm} Z_2^{\pm} \cdots Z_m^{\pm} H_m^{+},$$5 is separable, then $$X^{\pm} \approx Z_1^{\pm} Z_2^{\pm} \cdots Z_m^{\pm} H_m^{+},$$6 identifies $$X^{\pm} \approx Z_1^{\pm} Z_2^{\pm} \cdots Z_m^{\pm} H_m^{+},$$7 and $$X^{\pm} \approx Z_1^{\pm} Z_2^{\pm} \cdots Z_m^{\pm} H_m^{+},$$8 up to column permutations and scalings. In the neuroimaging interpretation, separability corresponds to each community having at least one anchor ROI that belongs exclusively to it (Seyedi et al., 13 May 2026). Regularized DMF supplies a different identifiability mechanism by selecting a unique end-to-end map almost everywhere through weight decay (Kamber et al., 28 Mar 2026).

A recurring misconception is that DMF is inherently less interpretable than graph autoencoders or GNNs. The literature often argues the opposite for constrained models: SD3MF avoids message passing and black-box embeddings, while SGMC preserves a deep linear structure even after adding spectral geometric regularization (Seyedi et al., 13 May 2026, Boyarski et al., 2019). This suggests that interpretability in DMF is not incidental; it is usually engineered through the factorization itself.

5. Application domains and empirical behavior

In recommendation and matrix completion, DMF appears in several forms. DeepFM combines an FM term with an MLP so that the FM component recovers MF on sparse one-hot inputs and the deep component adds higher-order nonlinear feature interactions (Zhang, 2022). NNMF replaces the inner product by a feed-forward network and, on MovieLens 100K, reports RMSE $$X^{\pm} \approx Z_1^{\pm} Z_2^{\pm} \cdots Z_m^{\pm} H_m^{+},$$9 for a 4-hidden-layer network, compared with $A_i^{(m)} \approx \Psi^{(m)} S_i \Psi^{(m)\top},$0 for BiasedMF and $A_i^{(m)} \approx \Psi^{(m)} S_i \Psi^{(m)\top},$1 for NTN; on MovieLens 1M, the same architecture reports RMSE $A_i^{(m)} \approx \Psi^{(m)} S_i \Psi^{(m)\top},$2 (Dziugaite et al., 2015). A different DMF model designed for extendability to unseen rows and columns reports on MovieLens 1M that DMF-D achieves RMSE $A_i^{(m)} \approx \Psi^{(m)} S_i \Psi^{(m)\top},$3 and MAE $A_i^{(m)} \approx \Psi^{(m)} S_i \Psi^{(m)\top},$4 in discrete matrix completion (Nguyen et al., 2018). In tag-aware personalized recommendation, HDMF integrates deep-semantic modeling, hybrid learning, and matrix factorization, achieving MAP $A_i^{(m)} \approx \Psi^{(m)} S_i \Psi^{(m)\top},$5 and MRR $A_i^{(m)} \approx \Psi^{(m)} S_i \Psi^{(m)\top},$6 on Delicious, with MAP $A_i^{(m)} \approx \Psi^{(m)} S_i \Psi^{(m)\top},$7 that of MF_sf and MRR $A_i^{(m)} \approx \Psi^{(m)} S_i \Psi^{(m)\top},$8 that of DSPR (Xu et al., 2017).

Multi-view clustering is a major unsupervised application. MVC-DMF-PA combines deep semi-NMF with partition alignment and reports, among other results, ACC $A_i^{(m)} \approx \Psi^{(m)} S_i \Psi^{(m)\top},$9 on BBC, ACC $\Psi^{(m)} = \prod_{\ell=1}^{L} W_\ell^{(m)}$0 on BBCSport, and ACC $\Psi^{(m)} = \prod_{\ell=1}^{L} W_\ell^{(m)}$1 on MSRCV1 (Zhang et al., 2021). DMFAW adds adaptive feature weighting and late fusion, and reports that it outperforms all baselines across Handwritten, Caltech101-7, Caltech101-all, BBCSport, BBC, and ORL; its dynamic feature-selection mechanism also improves runtime, for example on BBCSport from $\Psi^{(m)} = \prod_{\ell=1}^{L} W_\ell^{(m)}$2s to $\Psi^{(m)} = \prod_{\ell=1}^{L} W_\ell^{(m)}$3s and on BBC from $\Psi^{(m)} = \prod_{\ell=1}^{L} W_\ell^{(m)}$4s to $\Psi^{(m)} = \prod_{\ell=1}^{L} W_\ell^{(m)}$5s relative to a fixed-$\Psi^{(m)} = \prod_{\ell=1}^{L} W_\ell^{(m)}$6 variant (Khalafaoui et al., 2024).

In graph and brain-network analysis, DMF has been specialized to multimodal populations of graphs. SD3MF generalizes SNMTF from unsupervised single-graph clustering to supervised prediction over populations of multimodal graphs, and on multimodal connectome datasets it reports HIV ACC $\Psi^{(m)} = \prod_{\ell=1}^{L} W_\ell^{(m)}$7 and AUC $\Psi^{(m)} = \prod_{\ell=1}^{L} W_\ell^{(m)}$8, BP ACC $\Psi^{(m)} = \prod_{\ell=1}^{L} W_\ell^{(m)}$9 and AUC $S_i$0, and PPMI ACC $S_i$1 and AUC $S_i$2 (Seyedi et al., 13 May 2026).

In vision and robotics, DMF has been used for synchronization and averaging tasks. Rotation Synchronization via Deep Matrix Factorization formulates synchronization as low-rank completion of a block matrix and reports median angular errors often within $S_i$3 of the leading methods, and below $S_i$4 in most 1DSfM scenes, while remaining unsupervised (Tejus et al., 2023). Multiple Rotation Averaging with Constrained Reweighting Deep Matrix Factorization uses a symmetric low-rank architecture with a rank-3 bottleneck, spanning-tree-based edge filtering, iterative reweighting, and dynamic depth selection; experimental results on 1DSfM, ETH, and Stanford 3D validate the effectiveness of the method (Li et al., 2024).

In inverse problems, DMF has even been used in blind inverse light transport. “Computational Mirrors” formulates an observed video as $S_i$5, parameterizes both $S_i$6 and $S_i$7 by untrained CNNs in a one-off manner, and reports recognizable reconstructions of hidden motion that outperform classical NMF and adapted blind deconvolution baselines (Aittala et al., 2019).

6. Open problems and directions

Several limitations recur. The 2020 survey identifies open questions on selecting depth and inner ranks, choosing losses beyond the Frobenius norm, establishing identifiability and stability for deep constrained factorizations, designing scalable algorithms with convergence guarantees, and extending the framework to tensors and richer side information (Handschutter et al., 2020). These are still active problems rather than settled methodology.

Optimization remains a central difficulty. The nonconvex problem is often likely NP-hard, and practical methods rely on initialization, implicit regularization, or carefully designed block updates. Sparse deep NMF already notes only local optimality for the full model, while SD3MF states that the objective is highly non-convex and that training relies on implicit regularization via small learning rates and small-scale initialization (Guo et al., 2017, Seyedi et al., 13 May 2026).

Theory is also incomplete outside the deep linear regime. Exact sharpness formulas are now known for overparameterized DMF with squared-error loss, but extending such formulas to nonlinear activations or other losses is explicitly identified as nontrivial (Kamber et al., 30 Sep 2025). The regularized theory likewise leaves open whether analogous uniqueness or Hessian-structure results persist for losses beyond squared error or for regularizers beyond $S_i$8 weight decay (Kamber et al., 28 Mar 2026).

Application-specific extensions are equally visible. SD3MF identifies scalability, more expressive yet interpretable supervised heads, and temporal or longitudinal modeling as future directions (Seyedi et al., 13 May 2026). DMFAW explicitly notes that performance depends on view quality and points to robustness under noisy views as future work (Khalafaoui et al., 2024). In recommendation, the 2022 survey suggests that deep factorization remains especially compelling when nonlinear interactions, side-information fusion, and cold-start mitigation are required, but it also emphasizes the corresponding increase in parameters, tuning complexity, and risk of overfitting (Zhang, 2022).

Taken together, the literature suggests a stable core characterization: DMF is not a single algorithm but a family of multilayer factorization models in which depth is used to impose hierarchical structure on low-rank approximation. Its modern development spans deep linear products, constrained nonnegative and semi-nonnegative models, multimodal graph factorizations, neural interaction functions for recommendation, and application-specific architectures with explicit structural priors. The persistent research challenge is to preserve the advantages that made matrix factorization attractive in the first place—interpretability, compactness, and structural identifiability—while exploiting the optimization and representational benefits associated with depth.