Theorems on Positive Data: On the Uniqueness of NMF

Laurberg, Hans; Christensen, Mads Græsbøll; Plumbley, Mark D.; Hansen, Lars Kai; Jensen, Søren Holdt

doi:10.1155/2008/764206

Cited by 149 publications

(112 citation statements)

References 8 publications

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“…Thomas [23] and Chen [3] first gave different geometric interpretations of NMF, and stated the uniqueness problem in a geometric way. Donoho et al [24] and Laurberg et al [25] later provided uniqueness conditions by exploiting the aforementioned geometric viewpoint. The sufficient conditions they provided, however, require one of the two matrix factors to contain a scaled identity matrix-a particularly strict condition.…”

Section: N On-negative Matrix Factorization (Nmf) Is the Problem Of (mentioning

confidence: 99%

Non-Negative Matrix Factorization Revisited: Uniqueness and Algorithm for Symmetric Decomposition

Huang

Sidiropoulos

Swami

2014

IEEE Trans. Signal Process.

252

248

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Abstract-Non-negative matrix factorization (NMF) has found numerous applications, due to its ability to provide interpretable decompositions. Perhaps surprisingly, existing results regarding its uniqueness properties are rather limited, and there is much room for improvement in terms of algorithms as well. Uniqueness aspects of NMF are revisited here from a geometrical point of view. Both symmetric and asymmetric NMF are considered, the former being tantamount to element-wise non-negative square-root factorization of positive semidefinite matrices. New uniqueness results are derived, e.g., it is shown that a sufficient condition for uniqueness is that the conic hull of the latent factors is a superset of a particular second-order cone. Checking this condition is shown to be NP-complete; yet this and other results offer insights on the role of latent sparsity in this context. On the computational side, a new algorithm for symmetric NMF is proposed, which is very different from existing ones. It alternates between Procrustes rotation and projection onto the non-negative orthant to find a non-negative matrix close to the span of the dominant subspace. Simulation results show promising performance with respect to the state-of-art. Finally, the new algorithm is applied to a clustering problem for co-authorship data, yielding meaningful and interpretable results.

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Section: N On-negative Matrix Factorization (Nmf) Is the Problem Of (mentioning

confidence: 99%

Non-Negative Matrix Factorization Revisited: Uniqueness and Algorithm for Symmetric Decomposition

Huang

Sidiropoulos

Swami

2014

IEEE Trans. Signal Process.

252

248

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“…In many cases non-negativity constraints are enough to recover the original signals if the signals are sparse enough [19]. Moreover, results on the conditions for establishing the uniqueness of NMF rely on zero-groundedness of the signals [4], or determining binary closure patterns in the signal ensemble [20], which typically explains the success of sparse coding techniques [18]. Learning sparse factors is generally a good indicator for achieving good compression, much in the same way as integer sequences with many zeros are typically well compressed.…”

Section: Related Workmentioning

confidence: 99%

“…In general the derived factorization is non-unique [3,4] but satisfactory for most applications. Satisfaction is generally based on some distortion measure, e.g.…”

Section: Introductionmentioning

confidence: 99%

Learning and storing the parts of objects: IMF

Fréin¹

2014

2014 IEEE International Workshop on Machine Learning for Signal Processing (MLSP)

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“…The NMF solution is generally not unique, or exact. For every invertible A we have a potential factorization [18,9],…”

Section: Nonnegative Matrix Factorizationmentioning

confidence: 99%

Ghostbusters: A Parts-based NMF algorithm

Fréin¹

2013

24th IET Irish Signals and Systems Conference (ISSC 2013)

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An exact nonnegative matrix decomposition algorithm is proposed. This is achieved by 1) Taking a nonlinear approximation of a sparse real-valued dataset at a given tolerance-to-error constraint, ε; 2) Choosing an arbitrary lectic ordering on the rows or column entries; And, then 3) systematically applying a closure operator, so that all closures are selected. Assuming a nonnegative hierarchical closure structure (a Galois lattice) ensures the data has a unique ordered overcomplete dictionary representation. Parts-based constraints on these closures can then be used to specify and supervise the form of the solution. We illustrate that this approach outperforms NMF on two standard NMF datasets: it exhibits the properties described above; It is correct and exact.

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Theorems on Positive Data: On the Uniqueness of NMF

Cited by 149 publications

References 8 publications

Non-Negative Matrix Factorization Revisited: Uniqueness and Algorithm for Symmetric Decomposition

Non-Negative Matrix Factorization Revisited: Uniqueness and Algorithm for Symmetric Decomposition

Learning and storing the parts of objects: IMF

Ghostbusters: A Parts-based NMF algorithm

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