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The first one is the coherence problem (or coherent set approach), where groups of states are sought that evolve under the action of a stochastic transition matrix in a way maximally distinguishable from other groups. The second one is a low rank factorization approach for stochastic matrices, called direct Bayesian model reduction (DBMR), which estimates the low rank factors directly from observed data. We show that DBMR results in a low rank model that is a projection of the full model, and exploit this insight to infer bounds on a quantitative measure of coherence within the reduced model. Both approaches can be formulated as optimization problems, and we also prove a bound between their respective objectives. On a broader scope, this work relates the two classical loss functions of nonnegative matrix factorization, namely the Frobenius norm and the generalized Kullback\u2013Leibler divergence, and suggests new links between likelihood-based and projection-based estimation of probabilistic models.<\/jats:p>","DOI":"10.1007\/s00332-024-10091-x","type":"journal-article","created":{"date-parts":[[2024,10,30]],"date-time":"2024-10-30T17:25:43Z","timestamp":1730309143000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Coherent Set Identification Via Direct Low Rank Maximum Likelihood Estimation"],"prefix":"10.1007","volume":"35","author":[{"given":"Robert M.","family":"Polzin","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ilja","family":"Klebanov","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nikolas","family":"N\u00fcsken","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"P\u00e9ter","family":"Koltai","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,10,30]]},"reference":[{"issue":"2","key":"10091_CR1","doi-asserted-by":"crossref","DOI":"10.1103\/RevModPhys.89.025007","volume":"89","author":"H Aref","year":"2017","unstructured":"Aref, H., Blake, J.R., Budi\u0161i\u0107, M., Cardoso, S.S., Cartwright, J.H., Clercx, H.J., El Omari, K., Feudel, U., Golestanian, R., Gouillart, E., van Heijst, G.F., Krasnopolskaya, T.S., Guer, Y.L., MacKay, R.S., Meleshko, V.V., Metcalfe, G., Mezi\u0107, I., de Moura, A.P.S., Piro, O., Speetjens, M.F.M., Sturman, R., Thiffeault, J.-L., Tuval, I.: Frontiers of chaotic advection. 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