@article{coifman2005,
  title     = "Geometric diffusions as a tool for harmonic analysis and
               structure definition of data: diffusion maps",
  author    = "Coifman, R R and Lafon, S and Lee, A B and Maggioni, M and
               Nadler, B and Warner, F and Zucker, S W",
  abstract  = "We provide a framework for structural multiscale geometric
               organization of graphs and subsets of R(n). We use diffusion
               semigroups to generate multiscale geometries in order to
               organize and represent complex structures. We show that
               appropriately selected eigenfunctions or scaling functions of
               Markov matrices, which describe local transitions, lead to
               macroscopic descriptions at different scales. The process of
               iterating or diffusing the Markov matrix is seen as a
               generalization of some aspects of the Newtonian paradigm, in
               which local infinitesimal transitions of a system lead to global
               macroscopic descriptions by integration. We provide a unified
               view of ideas from data analysis, machine learning, and
               numerical analysis.",
  journal   = "Proc. Natl. Acad. Sci. U. S. A.",
  publisher = "Proceedings of the National Academy of Sciences",
  volume    =  102,
  number    =  21,
  pages     = "7426--7431",
  month     =  may,
  year      =  2005,
  language  = "en"
}

@article{haghverdi2015,
  title     = "Diffusion maps for high-dimensional single-cell analysis of
               differentiation data",
  author    = "Haghverdi, Laleh and Buettner, Florian and Theis, Fabian J",
  abstract  = "MOTIVATION: Single-cell technologies have recently gained
               popularity in cellular differentiation studies regarding their
               ability to resolve potential heterogeneities in cell
               populations. Analyzing such high-dimensional single-cell data
               has its own statistical and computational challenges. Popular
               multivariate approaches are based on data normalization,
               followed by dimension reduction and clustering to identify
               subgroups. However, in the case of cellular differentiation, we
               would not expect clear clusters to be present but instead expect
               the cells to follow continuous branching lineages. RESULTS:
               Here, we propose the use of diffusion maps to deal with the
               problem of defining differentiation trajectories. We adapt this
               method to single-cell data by adequate choice of kernel width
               and inclusion of uncertainties or missing measurement values,
               which enables the establishment of a pseudotemporal ordering of
               single cells in a high-dimensional gene expression space. We
               expect this output to reflect cell differentiation trajectories,
               where the data originates from intrinsic diffusion-like
               dynamics. Starting from a pluripotent stage, cells move smoothly
               within the transcriptional landscape towards more differentiated
               states with some stochasticity along their path. We demonstrate
               the robustness of our method with respect to extrinsic noise
               (e.g. measurement noise) and sampling density heterogeneities on
               simulated toy data as well as two single-cell quantitative
               polymerase chain reaction datasets (i.e. mouse haematopoietic
               stem cells and mouse embryonic stem cells) and an RNA-Seq data
               of human pre-implantation embryos. We show that diffusion maps
               perform considerably better than Principal Component Analysis
               and are advantageous over other techniques for non-linear
               dimension reduction such as t-distributed Stochastic Neighbour
               Embedding for preserving the global structures and
               pseudotemporal ordering of cells. AVAILABILITY AND
               IMPLEMENTATION: The Matlab implementation of diffusion maps for
               single-cell data is available at
               https://www.helmholtz-muenchen.de/icb/single-cell-diffusion-map.
               CONTACT: fbuettner.phys@gmail.com,
               fabian.theis@helmholtz-muenchen.de SUPPLEMENTARY INFORMATION:
               Supplementary data are available at Bioinformatics online.",
  journal   = "Bioinformatics",
  publisher = "Oxford University Press (OUP)",
  volume    =  31,
  number    =  18,
  pages     = "2989--2998",
  month     =  sep,
  year      =  2015,
  language  = "en"
}

@article{haghverdi2016,
  title    = "Diffusion pseudotime robustly reconstructs lineage branching",
  author   = "Haghverdi, Laleh and B{\"u}ttner, Maren and Wolf, F Alexander and
              Buettner, Florian and Theis, Fabian J",
  abstract = "The temporal order of differentiating cells is intrinsically
              encoded in their single-cell expression profiles. We describe an
              efficient way to robustly estimate this order according to
              diffusion pseudotime (DPT), which measures transitions between
              cells using diffusion-like random walks. Our DPT software
              implementations make it possible to reconstruct the developmental
              progression of cells and identify transient or metastable states,
              branching decisions and differentiation endpoints.",
  journal  = "Nat. Methods",
  volume   =  13,
  number   =  10,
  pages    = "845--848",
  month    =  oct,
  year     =  2016,
  language = "en"
}

@article{wolf2018,
  title    = "{SCANPY}: large-scale single-cell gene expression data analysis",
  author   = "Wolf, F Alexander and Angerer, Philipp and Theis, Fabian J",
  abstract = "SCANPY is a scalable toolkit for analyzing single-cell gene
              expression data. It includes methods for preprocessing,
              visualization, clustering, pseudotime and trajectory inference,
              differential expression testing, and simulation of gene
              regulatory networks. Its Python-based implementation efficiently
              deals with data sets of more than one million cells (
              https://github.com/theislab/Scanpy ). Along with SCANPY, we
              present ANNDATA, a generic class for handling annotated data
              matrices ( https://github.com/theislab/anndata ).",
  journal  = "Genome Biol.",
  volume   =  19,
  number   =  1,
  pages    = "15",
  month    =  feb,
  year     =  2018,
  keywords = "Bioinformatics; Clustering; Differential expression testing;
              Graph analysis; Machine learning; Pseudotemporal ordering;
              Scalability; Single-cell transcriptomics; Trajectory inference;
              Visualization",
  language = "en"
}
