Let us list our main results following our four main building blocks:
(1) Computational complexity: we proved NP-hardness of CLRMA with l1 and linfinity norms which are two important variants. We were also able to understand sparse CLRMA problems better using geometric intuition and characterize situations where the optimal solution is unique (up to permutation and scaling).
(2) Provably correct algorithms: We have designed a fast and provable correct algorithm for NMF under the separability condition. We have also developed algorithms for minimum-volume NMF that allows to solve NMF under weaker assumptions than the separability condition. We have also developed a provably correct algorithms for (i) simplex-structured matrix factorization, (ii) linear-quadratic NMF, and (iii) Convolutive NMF.
(3) Heuristics: We have developed several heuristic algorithms for the following CLRMA problems: NMF (using momentum/extrapolation), subspace clustering (using sampling and multilayer graphs), positive semidefinite factorization (that are particularly useful for extended formulations), sparse component analysis, dictionary-based factorizations, robust symmetric NMF, NMF based on beta-divergence with equality constraints, . More recently, we have developed general optimization tools that can be used to solve CLRMA. They are based on first-order and block coordinate descent methods with momentum, and are guaranteed to converge to stationary point.
Moreover, we have came up with a new application of CLRMA. We proposed a completely new model using factorization of stable matrices; namely A = (J-R)Q where J is antisymmetric (J^T=-J), R and Q are positive semidefinite. All previous work used directly the matrix A, trying to control its eigenvalues directly. With our formulation, a matrix is stable if and only if it admits a factorization of the form (J-R)Q with the above mentioned constraints. This work lead to many other results; e.g. for matrix pairs and positive-real systems.
(4) Applications: For all the algorithms mentioned above, we have used them on real-world applications; with a focus on document classification, audio source separation and hyperspectral unmixing. We have also developed some new models particularly well suited for applications: distributionally robust NMF, generalized separable NMF, and off-diagonal symmetric NMF, for which we have also proposed efficient algorithms.
