Theory
======
Given a point cloud :math:`X` in :math:`\mathbb{R}^d`, one can use the `Čech complex <https://en.wikipedia.org/wiki/%C4%8Cech_complex>`_ :math:`\mathrm{\check{C}ech}_{\bullet}(X)` and its homology groups as a proxy for the spatial information contained in :math:`X`. In many cases of interest, :math:`X` comprises heterogenous data, i.e., there exists a semantically meaningful colouring :math:`c:X \to \{0, \ldots, s\}`. In such cases one is often interested in capturing not just spatial information of :math:`X` but also of spatial relations among subtypes within the data :ref:`[3] <Stolz, B. et. al.>`. The benefit of the Čech complex is that it is *functorial* with respect to inclusions; one could consider the induced maps among the subcomplexes of :math:`\mathrm{\check{C}ech}_{\bullet}(X)` induced by inclusions among subsets of :math:`X`. These induced maps provide spatial relation information, but the drawback is that computing these inclusion maps and the induced maps on persistent homology is a computationally expensive procedure. 

The chromatic alpha complex of :math:`X` is a filtered simplicial complex :math:`\mathrm{ChrAlpha}_{\bullet}(X, c)` with vertex set :math:`X` that aims to address this drawback, defined and studied in :ref:`[1] <di Montesano et. al.>`. It is a generalisation of the alpha complex. 

#. *Functoriality*: In the Čech complex, every inclusion :math:`A \hookrightarrow B` of subsets of :math:`X` induces an inclusion of Čech complexes :math:`\mathrm{\check{C}ech}_{\bullet}(A) \hookrightarrow \mathrm{\check{C}ech}_{\bullet}(B)`. In the chromatic alpha complex, this functoriality holds for a select family of subsets of :math:`X`, namely those of the form  :math:`X_I := \chi^{-1}(I)` where :math:`I \subset \{0, \ldots, s\}`. To be more precise, let :math:`c_I` denote the restriction of :math:`c` to :math:`X_I`. Then :math:`\mathrm{ChrAlpha}_{\bullet}(X_I, c_I)` is the subcomplex of :math:`\mathrm{ChrAlpha}_{\bullet}(X, c)` generated by :math:`X_I`. 

#. *Recovery of spatial information*: For every subset :math:`I \subset \{0, \ldots, s\}` there is a filtration-preserving inclusion map :math:`\mathrm{ChrAlpha}_{\bullet}(X_I, c_I) \hookrightarrow \mathrm{\check{C}ech}_{\bullet}(X_I)` into the Čech complex which is a level-wise homotopy equivalence. In particular :math:`\mathrm{ChrAlpha}_{\bullet}(X, c) \simeq \mathrm{\check{C}ech}_{\bullet}(X)`, and the persistent homotopy type of the chromatic alpha complex does not depend on the colouring. This means that one can use the chromatic alpha complex as a substitute for the Čech for persistent homology calculations.  

#. *Computational efficiency*: The chromatic alpha complex typically has far fewer simplices than the Čech complex :ref:`[2] <Edelsbrunner H., Nikitenko A. and Reitzner M.>`. For example, if :math:`I = \{m\}` for some :math:`m \in \{0, \ldots s\}` then :math:`\mathrm{ChrAlpha}_{\bullet}(X_I, c_I)` is the `alpha complex <https://gudhi.inria.fr/doc/3.0.0/group__alpha__complex.html>`_ of the :math:`m`-coloured points. This means that the chromatic alpha complex has computational advantages over the Čech complex in persistent homology calculations when :math:`d` is small and the number of points is large.

#. The chromatic alpha complex includes as special cases the Čech complex when :math:`c` is the colouring that gives every point a different colour, and the alpha complex when :math:`c` gives every point the same colour. One can view the chromatic alpha complex as interpolating between the functorial properties of the Čech complex and the sparsity of the alpha complex while preserving overall spatial information.

References
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#. _`di Montesano et. al.`, "Persistent Homology of Chromatic Alpha Complexes".
   Online preprint available at `<https://arxiv.org/abs/2212.03128>`_. Accessed: 2023-02-28 22:07:23 UTC.

#. _`Edelsbrunner H., Nikitenko A. and Reitzner M.`, "Expected Sizes of Poisson-Delaunay Mosaics and Their Discrete Morse Functions" (2017) in Advances in Applied Probability, 49(3) pp. 745-767. DOI: `10.1017/apr.2017.20 <https://doi.org/10.1017/apr.2017.20>`_.

#. _`Stolz, B. et. al.`, "Relational persistent homology for multispecies data with application to the tumor microenvironment" (2023). Online preprint available at `<https://arxiv.org/abs/2308.06205>`_. Accessed: 2023-08-31 22:07:23 16:04:05 UTC. 

#. E. Welzl, "Smallest enclosing disks (balls and ellipsoids)," 
   in New Results and New Trends in Computer Science, H. Maurer, Ed., 
   in Lecture Notes in Computer Science. Berlin, Heidelberg: Springer, 
   1991, pp. 359–370. DOI: `10.1007/BFb0038202 <https://doi.org/10.1007/BFb0038202>`_.