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Non-negative Matrix Factorization
Matrix decomposition algorithm that factorizes a matrix V into two non-negative matrices W and H such that V ≈ WH. This non-negativity constraint allows for an additive interpretation of the components, making the results more easily interpretable.
Basis Matrix
The matrix W in the NMF decomposition (V ≈ WH) containing the basis vectors or prototypes that represent the fundamental features extracted from the original data. Each column of this matrix captures a pattern or latent feature present in the data.
Coefficient Matrix
The matrix H in the NMF decomposition (V ≈ WH) containing the weights or activation coefficients that indicate how each basis feature contributes to reconstructing the original data. These coefficients allow each sample to be represented as an additive combination of the basis features.
Multiplicative Update Algorithm
An iterative optimization algorithm specific to NMF that updates the matrices W and H using multiplicative update rules ensuring non-negativity. This algorithm alternately minimizes the matrices W and H while keeping their elements non-negative.
Reconstruction Cost
A quantitative measure of the error between the original matrix V and its factorization WH, usually calculated as the Frobenius distance or KL divergence. This metric guides the optimization process and evaluates the quality of the obtained decomposition.
Latent Features
Unobserved underlying components discovered by NMF from the raw data, representing intrinsic patterns or structures. These features emerge naturally from the decomposition and reveal hidden relationships in the data.
Sparsity
A desirable property in NMF where most of the elements in the W or H matrices are zero or close to zero, favoring more interpretable representations. The sparsity constraint helps to isolate the most relevant features and to reduce redundancy.
Orthogonal NMF
A variant of NMF that imposes an orthogonality constraint on the coefficient matrix H, promoting a clear partitioning of the data into distinct clusters. This approach improves feature separation and facilitates the interpretation of results in clustering tasks.
Sparse NMF
Extension of NMF incorporating additional sparsity constraints on the W or H matrices to obtain more sparse representations. This method is particularly effective for feature selection and identifying the most significant components.
Convex NMF
Variant of NMF where the basis vectors are constrained to be convex combinations of the original samples, improving the interpretability of the results. This approach ensures that each discovered feature can be expressed as a weighted average of the input data.
Alternating iterations
Optimization strategy in NMF where the W and H matrices are updated alternately, with one being fixed while the other is optimized. This approach guarantees convergence to a locally optimal solution while maintaining the non-negativity constraints.
Local convergence
Property of NMF algorithms that converge to a local rather than global optimum due to the non-convexity of the problem. The quality of the final solution often depends on the initialization and may require multiple runs to find the best decomposition.
Random initialization
Common method for starting the NMF algorithm by randomly initializing the W and H matrices with non-negative values. The initialization significantly influences the convergence and the quality of the final solution obtained after optimization.
NMF for classification
Application of NMF as a dimensionality reduction technique where the coefficients of the H matrix are used as features for classification algorithms. This approach often improves classification performance by providing more discriminative representations.
NMF for clustering
Use of NMF to discover a natural clustering structure in the data by interpreting the coefficients of the H matrix as cluster memberships. Each sample is assigned to the cluster corresponding to the maximum coefficient in its representation.
Probabilistic NMF
Probabilistic framework for NMF that models the data with statistical distributions like Poisson or Gaussian, offering a Bayesian interpretation of the factorization. This approach allows for integrating prior knowledge and quantifying the uncertainty of the estimates.
Regularized NMF
Extension of NMF incorporating regularization terms in the objective function to control model complexity and avoid overfitting. Regularization can impose sparsity, smoothness, or other desirable constraints on the factorized matrices.