Least squares algorithms under unimodality and non-negativity constraints

Rasmus Bro, Nicholaos D. Sidiropoulos

Research output: Contribution to journalArticlepeer-review

138 Scopus citations


In this paper a least squares method is developed for minimizing ||Y - XBT||2F over the matrix B subject to the constraint that the columns of B are unimodal, i.e. each has only one peak, and ||M||2F being the sum of squares of all elements of M. This method is directly applicable in many curve resolution problems, but also for stabilizing other problems where unimodality is known to be a valid assumption. Typical problems arise in certain types of time series analysis such as chromatography or flow injection analysis. A fundamental and surprising result of this work is that unimodal least squares regression (including optimization of mode location) is not any more difficult than two simple Kruskal monotone regressions. This had not been realized earlier, leading to the use of either undesirable ad hoc methods or very time-consuming exhaustive search algorithms. The new method is useful in and exemplified with two- and multi-way methods based on alternating least squares regression solving problems from fluorescence spectroscopy and flow injection analysis.

Original languageEnglish (US)
Pages (from-to)223-247
Number of pages25
JournalJournal of Chemometrics
Issue number4
StatePublished - Jan 1 1998


  • Alternating least squares regression
  • Chromatography
  • Curve resolution
  • Flow injection analysis
  • Fluorescence spectroscopy
  • Non-negativity
  • Oligomodal
  • Unimodal least squares regression


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