simplicial cone
State Aggregation Learning from Markov Transition Data
Yaqi Duan, Tracy Ke, Mengdi Wang
Inthispaper,wepropose a tractable algorithm that estimates the probabilistic aggregation map from the system'strajectory.
State Aggregation Learning from Markov Transition Data
Yaqi Duan, Tracy Ke, Mengdi Wang
State aggregation is a popular model reduction method rooted in optimal control. It reduces the complexity of engineering systems by mapping the system's states
A Characterization of the Non-Uniqueness of Nonnegative Matrix Factorizations
Nonnegative matrix factorization (NMF) is a popular dimension reduction technique that produces interpretable decomposition of the data into parts. However, this decompostion is not generally identifiable (even up to permutation and scaling). While other studies have provide criteria under which NMF is identifiable, we present the first (to our knowledge) characterization of the non-identifiability of NMF. We describe exactly when and how non-uniqueness can occur, which has important implications for algorithms to efficiently discover alternate solutions, if they exist.
When Does Non-Negative Matrix Factorization Give a Correct Decomposition into Parts?
Donoho, David, Stodden, Victoria
We interpret nonnegative matrix factorization geometrically, as the problem of finding a simplicial cone which contains a cloud of data points and which is contained in the positive orthant. We show that under certain conditions, basically requiring that some of the data are spread across the faces of the positive orthant, there is a unique such simplicial cone. We give examples of synthetic image articulation databases which obey these conditions; these require separated support and factorial sampling. For such databases there is a generative model in terms of'parts' and NMF correctly identifies the'parts'. We show that our theoretical results are predictive of the performance of published NMF code, by running the published algorithms on one of our synthetic image articulation databases.
When Does Non-Negative Matrix Factorization Give a Correct Decomposition into Parts?
Donoho, David, Stodden, Victoria
We interpret nonnegative matrix factorization geometrically, as the problem of finding a simplicial cone which contains a cloud of data points and which is contained in the positive orthant. We show that under certain conditions, basically requiring that some of the data are spread across the faces of the positive orthant, there is a unique such simplicial cone. We give examples of synthetic image articulation databases which obey these conditions; these require separated support and factorial sampling. For such databases there is a generative model in terms of'parts' and NMF correctly identifies the'parts'. We show that our theoretical results are predictive of the performance of published NMF code, by running the published algorithms on one of our synthetic image articulation databases.
When Does Non-Negative Matrix Factorization Give a Correct Decomposition into Parts?
Donoho, David, Stodden, Victoria
We interpret nonnegative matrix factorization geometrically, as the problem of finding a simplicial cone which contains a cloud of data points and which is contained in the positive orthant. We show that under certain conditions, basically requiring that some of the data are spread across the faces of the positive orthant, there is a unique such simplicial cone.We give examples of synthetic image articulation databases which obey these conditions; these require separated support and factorial sampling.For such databases there is a generative model in terms of'parts' and NMF correctly identifies the'parts'. We show that our theoretical results are predictive of the performance of published NMF code, by running the published algorithms on one of our synthetic image articulation databases.