Partially Fixed Nonnegative Matrix Factorization

• Partially fixed NMF is a variant that incorporates prior knowledge by fixing complete factors or selected entries, thereby improving interpretability and identifiability.
• It utilizes fixed-factor, group-restricted, and data-anchored formulations to transform nonconvex problems into convex subproblems in specific settings.
• Optimization methods like multiplicative updates and proximal algorithms are employed to efficiently solve these models under noise and missing data constraints.

Searching arXiv for additional relevant papers on partially fixed and constrained NMF. Partially fixed nonnegative matrix factorization denotes a family of NMF models in which the factorization is not fully free: one factor may be fixed, selected rows or columns of a factor may be prescribed, known group structure may be injected on the coefficient side, or parts of the decomposition may be anchored to rows or columns of the data matrix itself. Relative to standard NMF, which approximates a nonnegative matrix by a product of nonnegative factors, these models are motivated by settings in which some latent basis vectors, some observation memberships, or some factor components are already known and the remaining latent structure must be estimated conditionally on that prior information (Shreeves et al., 2021, Yanez et al., 2014, Pan et al., 2019). In exact-factorization settings, the same topic also intersects partial identifiability, where the central question is not whether the entire factorization is unique, but whether specific columns of a factor are uniquely forced by the data (Gillis et al., 2022).

1. Standard NMF and the rationale for partial fixation

Standard NMF starts from a nonnegative matrix and seeks lower-rank nonnegative factors. In the notation used by "Nonnegative Matrix Factorization with Group and Basis Restrictions" (Shreeves et al., 2021), the model is

$\mathbf{X} \approx \mathbf{W}\mathbf{H},$

or, with noise,

$$\mathbf{X} = \mathbf{W}\mathbf{H} + \boldsymbol{\varepsilon},$$

with $\mathbf{X} \in \mathbb{R}_{\ge 0}^{n \times p}$, $\mathbf{W} \in \mathbb{R}_{\ge 0}^{n \times q}$, and $\mathbf{H} \in \mathbb{R}_{\ge 0}^{q \times p}$. Under Gaussian noise, the associated objective is the Frobenius-loss criterion

$$D_F(\mathbf{X}\mid \mathbf{W}\mathbf{H}) = \frac{1}{2}\|\mathbf{X}-\mathbf{W}\mathbf{H}\|_F^2.$$

In the KL-divergence setting used by "Primal-Dual Algorithms for Non-negative Matrix Factorization with the Kullback-Leibler Divergence" (Yanez et al., 2014), the model is

$V \approx WH,$

with the full optimization problem

$\min_{W,H\ge 0} D(V\|WH),$

where $D(\cdot\|\cdot)$ is the generalized KL divergence.

The principal motivation for partially fixed NMF is that fully unsupervised NMF may fail to recover factors already known to exist a priori, or may produce factors that are less interpretable than desired. The most common forms of prior information described in the cited literature are known or suspected latent basis vectors, known group memberships of some observations, and settings in which one factor acts as a known nonnegative dictionary while the other is to be estimated (Shreeves et al., 2021, Yanez et al., 2014). In this sense, partially fixed NMF is less a single algorithm than a problem class spanning fixed-basis NMF, one-factor-fixed nonnegative decomposition, semi-supervised NMF, guided NMF, constrained NMF, and data-anchored separable or generalized separable factorizations.

A basic structural distinction runs through the literature. In one class of models, one factor is fixed entirely and the remaining factor is estimated. In another, only selected rows, columns, or entries are held fixed. In a third, the fixed structure is not externally supplied but is required to coincide with rows or columns of the data matrix itself. These distinctions govern both the optimization geometry and the type of guarantees available.

2. Canonical formulations of partially fixed NMF

The simplest partially fixed formulation is the one-factor-fixed nonnegative decomposition problem. Because the full KL-NMF objective is nonconvex jointly in $(W,H)$ but convex in each factor separately, fixing one factor yields a convex subproblem: $$\mathbf{X} = \mathbf{W}\mathbf{H} + \boldsymbol{\varepsilon},$$0 Columnwise, this reduces to

$$\mathbf{X} = \mathbf{W}\mathbf{H} + \boldsymbol{\varepsilon},$$1

where $$\mathbf{X} = \mathbf{W}\mathbf{H} + \boldsymbol{\varepsilon},$$2 is known, $$\mathbf{X} = \mathbf{W}\mathbf{H} + \boldsymbol{\varepsilon},$$3 is fixed and nonnegative, and $$\mathbf{X} = \mathbf{W}\mathbf{H} + \boldsymbol{\varepsilon},$$4 is the unknown nonnegative coefficient vector (Yanez et al., 2014). This is the most literal fixed-factor instance of partially fixed NMF.

A more elaborate basis-restricted construction is Group and Basis Restricted NMF (GBR-NMF), introduced in "Nonnegative Matrix Factorization with Group and Basis Restrictions" (Shreeves et al., 2021). Its central model is

$$\mathbf{X} = \mathbf{W}\mathbf{H} + \boldsymbol{\varepsilon},$$5

with $$\mathbf{X} = \mathbf{W}\mathbf{H} + \boldsymbol{\varepsilon},$$6, $$\mathbf{X} = \mathbf{W}\mathbf{H} + \boldsymbol{\varepsilon},$$7, and $$\mathbf{X} = \mathbf{W}\mathbf{H} + \boldsymbol{\varepsilon},$$8. The first $$\mathbf{X} = \mathbf{W}\mathbf{H} + \boldsymbol{\varepsilon},$$9 columns of $\mathbf{X} \in \mathbb{R}_{\ge 0}^{n \times p}$0 may be fixed to a known grouping matrix, selected rows of $\mathbf{X} \in \mathbb{R}_{\ge 0}^{n \times p}$1 may be fixed to known basis vectors, and the remaining components remain free, with the compatibility condition $\mathbf{X} \in \mathbb{R}_{\ge 0}^{n \times p}$2 when both group and basis restrictions are used. The key technical device is the auxiliary matrix $\mathbf{X} \in \mathbb{R}_{\ge 0}^{n \times p}$3, initialized as $\mathbf{X} \in \mathbb{R}_{\ge 0}^{n \times p}$4, which is introduced to “perform adjustments on the constrained factors.” This turns hard fixed-basis NMF into a restricted basis model in which the effective factor matrix is modulated through $\mathbf{X} \in \mathbb{R}_{\ge 0}^{n \times p}$5.

A different axis of partial fixation appears in generalized separable NMF. In "Generalized Separable Nonnegative Matrix Factorization" (Pan et al., 2019), a matrix $\mathbf{X} \in \mathbb{R}_{\ge 0}^{n \times p}$6 is $\mathbf{X} \in \mathbb{R}_{\ge 0}^{n \times p}$7-separable if there exist index sets $\mathbf{X} \in \mathbb{R}_{\ge 0}^{n \times p}$8 and $\mathbf{X} \in \mathbb{R}_{\ge 0}^{n \times p}$9 such that

$\mathbf{W} \in \mathbb{R}_{\ge 0}^{n \times q}$0

with

$\mathbf{W} \in \mathbb{R}_{\ge 0}^{n \times q}$1

Here partial fixation is data-driven rather than externally imposed: some components are anchored as columns of the data matrix and others as rows of the data matrix. The equivalent self-dictionary form

$\mathbf{W} \in \mathbb{R}_{\ge 0}^{n \times q}$2

makes this a structured selection problem with sparse rows in $\mathbf{W} \in \mathbb{R}_{\ge 0}^{n \times q}$3 and sparse columns in $\mathbf{W} \in \mathbb{R}_{\ge 0}^{n \times q}$4.

When data are incomplete, the masked least-squares formulation used in "An Alternating Direction Algorithm for Matrix Completion with Nonnegative Factors" (Xu et al., 2011) provides another natural envelope for partial fixation: $\mathbf{W} \in \mathbb{R}_{\ge 0}^{n \times q}$5 Although that paper focuses on nonnegative matrix factorization/completion, its split formulation separates bilinear fitting, observation consistency, and factor-side constraints, so the same template is directly relevant when one factor, selected blocks, or selected entries are fixed.

3. Optimization methods and update mechanisms

For GBR-NMF, the objective under Gaussian error is

$\mathbf{W} \in \mathbb{R}_{\ge 0}^{n \times q}$6

subject to nonnegativity and hard partial-fixing constraints. The paper derives Lee–Seung-style multiplicative updates for unconstrained entries of $\mathbf{W} \in \mathbb{R}_{\ge 0}^{n \times q}$7, all entries of $\mathbf{W} \in \mathbb{R}_{\ge 0}^{n \times q}$8, and unconstrained entries of $\mathbf{W} \in \mathbb{R}_{\ge 0}^{n \times q}$9, while specified columns of $\mathbf{H} \in \mathbb{R}_{\ge 0}^{q \times p}$0 and specified rows of $\mathbf{H} \in \mathbb{R}_{\ge 0}^{q \times p}$1 are held fixed. The algorithm initializes fixed columns of $\mathbf{H} \in \mathbb{R}_{\ge 0}^{q \times p}$2 with the known cluster membership matrix, initializes $\mathbf{H} \in \mathbb{R}_{\ge 0}^{q \times p}$3 as identity, initializes unknown rows of $\mathbf{H} \in \mathbb{R}_{\ge 0}^{q \times p}$4 randomly, then alternates block updates until the improvement falls below $\mathbf{H} \in \mathbb{R}_{\ge 0}^{q \times p}$5 or a maximum iteration count is reached; the paper usually uses a large maximum iteration count of $\mathbf{H} \in \mathbb{R}_{\ge 0}^{q \times p}$6. A standard auxiliary-function argument is given to show that the objective is non-increasing under the updates and invariant only at stationary points, so convergence is to a local stationary point rather than a global optimum (Shreeves et al., 2021).

For fixed-factor KL problems, "Primal-Dual Algorithms for Non-negative Matrix Factorization with the Kullback-Leibler Divergence" (Yanez et al., 2014) reformulates

$\mathbf{H} \in \mathbb{R}_{\ge 0}^{q \times p}$7

as

$\mathbf{H} \in \mathbb{R}_{\ge 0}^{q \times p}$8

derives the Fenchel dual, and applies the Chambolle–Pock algorithm. The resulting proximal operators are available in closed form: $\mathbf{H} \in \mathbb{R}_{\ge 0}^{q \times p}$9 This yields a first-order primal-dual solver using only applications of $$D_F(\mathbf{X}\mid \mathbf{W}\mathbf{H}) = \frac{1}{2}\|\mathbf{X}-\mathbf{W}\mathbf{H}\|_F^2.$$0, $$D_F(\mathbf{X}\mid \mathbf{W}\mathbf{H}) = \frac{1}{2}\|\mathbf{X}-\mathbf{W}\mathbf{H}\|_F^2.$$1, projection onto $$D_F(\mathbf{X}\mid \mathbf{W}\mathbf{H}) = \frac{1}{2}\|\mathbf{X}-\mathbf{W}\mathbf{H}\|_F^2.$$2, and elementwise operations. For the fixed-factor subproblem, the method inherits convex convergence theory, strong duality, and a duality-gap certificate; when used in alternating NMF, however, only the subproblem solves are covered by the convex theory.

In the missing-data setting, the alternating-direction augmented-Lagrangian scheme of (Xu et al., 2011) introduces auxiliary variables $$D_F(\mathbf{X}\mid \mathbf{W}\mathbf{H}) = \frac{1}{2}\|\mathbf{X}-\mathbf{W}\mathbf{H}\|_F^2.$$3 and solves

$$D_F(\mathbf{X}\mid \mathbf{W}\mathbf{H}) = \frac{1}{2}\|\mathbf{X}-\mathbf{W}\mathbf{H}\|_F^2.$$4

The $$D_F(\mathbf{X}\mid \mathbf{W}\mathbf{H}) = \frac{1}{2}\|\mathbf{X}-\mathbf{W}\mathbf{H}\|_F^2.$$5- and $$D_F(\mathbf{X}\mid \mathbf{W}\mathbf{H}) = \frac{1}{2}\|\mathbf{X}-\mathbf{W}\mathbf{H}\|_F^2.$$6-updates are closed-form linear solves involving only $$D_F(\mathbf{X}\mid \mathbf{W}\mathbf{H}) = \frac{1}{2}\|\mathbf{X}-\mathbf{W}\mathbf{H}\|_F^2.$$7 inverses, the $$D_F(\mathbf{X}\mid \mathbf{W}\mathbf{H}) = \frac{1}{2}\|\mathbf{X}-\mathbf{W}\mathbf{H}\|_F^2.$$8-update overwrites observed entries with the data, and the $$D_F(\mathbf{X}\mid \mathbf{W}\mathbf{H}) = \frac{1}{2}\|\mathbf{X}-\mathbf{W}\mathbf{H}\|_F^2.$$9 updates are projections onto the nonnegative orthant. The paper treats this as an algorithmic template rather than a theory of partial fixation, but its variable splitting isolates the exact places where hard equality constraints can be imposed.

For generalized separable NMF, the main optimization model is the penalized convex program

$V \approx WH,$0

optimized over convex sets $V \approx WH,$1 that enforce nonnegativity and diagonal-dominance constraints. The associated GS-FGM algorithm is a projected accelerated gradient method with per-iteration complexity $V \approx WH,$2, while GSPA is a cheaper greedy heuristic with complexity $V \approx WH,$3 (Pan et al., 2019).

4. Identifiability, uniqueness, and structural guarantees

A central theoretical issue for partially fixed NMF is whether a fixed or recovered component is genuinely forced by the data. "Partial Identifiability for Nonnegative Matrix Factorization" (Gillis et al., 2022) formalizes this in Exact NMF. For an exact factorization

$V \approx WH,$4

the $V \approx WH,$5-th column of $V \approx WH,$6 is identifiable if every other Exact NMF of $V \approx WH,$7 contains that column up to positive scaling. This is a columnwise uniqueness notion, distinct from full identifiability of the entire factor pair.

The paper proves a restricted DBU theorem. Let

$V \approx WH,$8

If

$V \approx WH,$9

and there exists a row $\min_{W,H\ge 0} D(V\|WH),$0 of $\min_{W,H\ge 0} D(V\|WH),$1 such that

$\min_{W,H\ge 0} D(V\|WH),$2

then the $\min_{W,H\ge 0} D(V\|WH),$3-th column of $\min_{W,H\ge 0} D(V\|WH),$4 is identifiable. The same paper also gives a geometric reinterpretation in terms of minimal faces of the outer polytope, a broader geometric identifiability theorem based on face disjointness, a stronger theorem for $\min_{W,H\ge 0} D(V\|WH),$5, and a sequential theorem showing how already identifiable columns can be used to reduce the problem and certify additional columns. These results do not prescribe an optimization algorithm with hard fixed columns, but they supply the exact uniqueness logic that can justify fixing or trusting specific components.

A second line of theory concerns making NMF more sparse and more well-posed before any factor-side fixation is imposed. "Sparse and Unique Nonnegative Matrix Factorization Through Data Preprocessing" (Gillis, 2012) introduces a preprocessing operator

$\min_{W,H\ge 0} D(V\|WH),$6

where $\min_{W,H\ge 0} D(V\|WH),$7 is inverse-positive, with the goal of preserving recoverability while increasing sparsity and tightening the geometry of the factorization problem. The paper proves that, if no column of $\min_{W,H\ge 0} D(V\|WH),$8 is a multiple of another, then $\min_{W,H\ge 0} D(V\|WH),$9, $D(\cdot\|\cdot)$0, and $D(\cdot\|\cdot)$1 is inverse-positive. Under separability, preprocessing is provably optimal, and for rank-three matrices it can make the number of exact factorizations finite. This is not a partially fixed NMF theory, but it is directly relevant to the conditioning of the free part of a partially fixed model.

In near-separable settings, "Robust Near-Separable Nonnegative Matrix Factorization Using Linear Optimization" (Gillis et al., 2013) shifts attention from arbitrary factor matrices to anchor selection. Its LP uses a localization matrix $D(\cdot\|\cdot)$2 whose diagonal indicates whether a data column is selected as an anchor, with constraints of the form

$D(\cdot\|\cdot)$3

The paper does not formulate a generic partially fixed NMF model, but it explicitly describes how the LP can be adapted when some anchor indices are known in advance by enforcing fixed diagonal values, and likewise when some columns are known not to be anchors. This is a highly specific but practically important form of partial fixation.

5. Representative model classes and applications

GBR-NMF provides a direct demonstration of simultaneous factor-side restrictions in both $D(\cdot\|\cdot)$4 and $D(\cdot\|\cdot)$5. In the simulated study of (Shreeves et al., 2021), the setup uses $D(\cdot\|\cdot)$6, $D(\cdot\|\cdot)$7, $D(\cdot\|\cdot)$8 groups, and $D(\cdot\|\cdot)$9 true factors; the first four columns of $(W,H)$0 are fixed to a known classification matrix and one common factor in $(W,H)$1 is held fixed. The reported outcome is better recovery of the true factors and scores than standard NMF, with notably improved RSS comparisons.

The same paper gives two concrete real-data examples. For facial expression data, the dataset contains 84 facial images of 12 women with 7 facial expressions; known facial-expression groups are encoded as constraints in $(W,H)$2, while neutral faces are constrained as basis information in $(W,H)$3. GBR-NMF recovered facial features more aligned with expression-specific structures, whereas standard NMF tended to recover person-specific patterns. For Raman spectroscopy, the data contain 3240 spectra, 582 Raman intensity variables, and 3 known groups corresponding to the cell lines H460, MCF-7, and LNCaP; a known glycogen spectrum is fixed as a row of $(W,H)$4, known cell-line labels constrain columns of $(W,H)$5, and one additional component is left fully unconstrained. The paper reports that this preserves a biologically validated glycogen factor while revealing a new unconstrained spectral component that, together with glycogen scores, separates prostate cells from the other lines.

The one-factor-fixed KL setting of (Yanez et al., 2014) is aligned with applications in which a nonnegative dictionary is known and activations must be estimated. The paper discusses synthetic data, face recognition, and music source separation, and emphasizes that KL loss is particularly important for nonnegative count-like or intensity-like data such as spectrogram magnitudes in audio, document-word counts or frequencies, image intensities, and Poisson-like observations.

Generalized separable NMF produces mixed row/column anchoring rather than externally supplied fixed factors. On synthetic $(W,H)$6-separable $(W,H)$7 matrices with noise, GS-FGM recovers the correct row and column anchors very accurately and often exactly up to substantial noise, while on facial image datasets the selected columns correspond to representative faces and the selected rows to representative pixels (Pan et al., 2019). Although this is not the same as prescribing factors externally, it demonstrates that partial fixation can operate on both sides of the factorization.

In incomplete-data settings, the NMFC framework of (Xu et al., 2011) shows that nonnegativity and low-rank factorization can be exploited jointly when only a subset of entries is observed. The paper reports that, on tasks of recovering incomplete grayscale and hyperspectral images, the proposed algorithm yields overall better qualities than two recent matrix-completion algorithms that do not exploit nonnegativity. This is relevant because partially fixed NMF is often required precisely when prior components must be imposed in the presence of missing observations.

6. Limitations, misconceptions, and methodological boundaries

The cited literature is consistent on several limitations. First, prior knowledge must be correct. In GBR-NMF, if the supplied groups or basis rows are wrong, the method may force the factorization in an unhelpful direction; moreover, the restriction form is specific, since the prior information must be expressible as fixed columns of $(W,H)$8 and/or fixed rows of $(W,H)$9 (Shreeves et al., 2021). Partially fixed NMF is therefore not an arbitrary constrained-NMF framework by default.

Second, optimization guarantees are usually blockwise or local. GBR-NMF is nonconvex jointly in $$\mathbf{X} = \mathbf{W}\mathbf{H} + \boldsymbol{\varepsilon},$$00, and its multiplicative updates converge only to local stationary points. In the KL setting, the fixed-factor problem is convex and admits strong duality and a duality-gap certificate, but once one alternates between the two factor blocks the full problem becomes nonconvex again (Yanez et al., 2014). The missing-data augmented-Lagrangian algorithm similarly offers a KKT-type accumulation-point result under additional assumptions rather than a global convergence theorem (Xu et al., 2011).

Third, exact uniqueness theory does not automatically transfer to noisy or approximate settings. The partial identifiability results of (Gillis et al., 2022) assume Exact NMF, the rank condition $$\mathbf{X} = \mathbf{W}\mathbf{H} + \boldsymbol{\varepsilon},$$01, and typically a selective window condition in which a data column equals the target component up to scaling. The paper is explicit that it does not provide a theoretical extension to approximate NMF. Likewise, the strongest preprocessing guarantees in (Gillis, 2012) concern exact or separable settings, and the paper explicitly states that it does not analyze models with fixed columns or entries of the factors.

Fourth, data-anchored models solve a narrower problem than generic partially fixed NMF. Near-separable LP formulations and generalized separable NMF assume that unknown basis atoms are columns or rows of the data matrix. This is well matched to pure-pixel, anchor-word, and self-dictionary settings, but it is less directly applicable when the fixed basis vectors are arbitrary external templates. The GS-NMF paper also stresses that generalized separable decompositions are generally not unique, even though it provides a uniqueness condition under additional zero-pattern assumptions (Pan et al., 2019).

Taken together, these results suggest a precise interpretation of partially fixed NMF. It is not merely NMF with ad hoc constraints, but a spectrum of models in which interpretability, prior information, and identifiability are traded against flexibility. The strongest direct formulations are those in which one factor, selected rows or columns, or selected group indicators can be written explicitly into the optimization problem; the strongest theoretical guarantees arise in exact or separable regimes; and the most practically effective methods are those that align the form of the fixed structure with the geometry of the data.