Solving systems of polynomial inequalities in subexponential time
Journal of Symbolic Computation
Expressing combinatorial optimization problems by linear programs
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
Lower bounds for non-commutative computation
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
Communication complexity and combinatorial lattice theory
Journal of Computer and System Sciences
On the combinatorial and algebraic complexity of quantifier elimination
Journal of the ACM (JACM)
Complexity and real computation
Complexity and real computation
Discrete Applied Mathematics
Latent semantic indexing: a probabilistic analysis
Journal of Computer and System Sciences - Special issue on the seventeenth ACM SIGACT-SIGMOD-SIGART symposium on principles of database systems
Journal of Computer and System Sciences - Special issue on the fourteenth annual IEE conference on computational complexity
Recommendation systems: a probabilistic analysis
Journal of Computer and System Sciences - Special issue on Internet algorithms
Lectures on Discrete Geometry
On notions of information transfer in VLSI circuits
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
Document clustering based on non-negative matrix factorization
Proceedings of the 26th annual international ACM SIGIR conference on Research and development in informaion retrieval
The Journal of Machine Learning Research
Non-negative Matrix Factorization with Sparseness Constraints
The Journal of Machine Learning Research
Using mixture models for collaborative filtering
Journal of Computer and System Sciences
On the possibility of faster SAT algorithms
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
On the Complexity of Nonnegative Matrix Factorization
SIAM Journal on Optimization
On the number of separable partitions
Journal of Combinatorial Optimization
Probabilistic latent semantic analysis
UAI'99 Proceedings of the Fifteenth conference on Uncertainty in artificial intelligence
Large-scale distributed non-negative sparse coding and sparse dictionary learning
Proceedings of the 18th ACM SIGKDD international conference on Knowledge discovery and data mining
Critical point methods and effective real algebraic geometry: new results and trends
Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation
An information complexity approach to extended formulations
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Sparse and unique nonnegative matrix factorization through data preprocessing
The Journal of Machine Learning Research
Most Tensor Problems Are NP-Hard
Journal of the ACM (JACM)
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The Nonnegative Matrix Factorization (NMF) problem has a rich history spanning quantum mechanics, probability theory, data analysis, polyhedral combinatorics, communication complexity, demography, chemometrics, etc. In the past decade NMF has become enormously popular in machine learning, where the factorization is computed using a variety of local search heuristics. Vavasis recently proved that this problem is NP-complete. We initiate a study of when this problem is solvable in polynomial time. Consider a nonnegative m x n matrix $M$ and a target inner-dimension r. Our results are the following: - We give a polynomial-time algorithm for exact and approximate NMF for every constant r. Indeed NMF is most interesting in applications precisely when r is small. We complement this with a hardness result, that if exact NMF can be solved in time (nm)o(r), 3-SAT has a sub-exponential time algorithm. Hence, substantial improvements to the above algorithm are unlikely. - We give an algorithm that runs in time polynomial in n, m and r under the separablity condition identified by Donoho and Stodden in 2003. The algorithm may be practical since it is simple and noise tolerant (under benign assumptions). Separability is believed to hold in many practical settings. To the best of our knowledge, this last result is the first polynomial-time algorithm that provably works under a non-trivial condition on the input matrix and we believe that this will be an interesting and important direction for future work.