<span style="font-variant:small-caps;">mri</span> tractographs, movie clips), text data (e.g. blogs, tweets), etc., often come in the form of a set of feature vectors $a_1,\dots,a_m \in \mathbb{R}^d$ and can be conveniently represented by a matrix $A \in \mathbb{R}^{m \times d}$ (e.g. gene-microarray matrices of gene expression levels, frame-pixel matrices of grey scale values, term-document matrices of term frequencies-inverse document frequencies). In modern applications, it is often the case that one will encounter an exceedingly large sample size $m$ (massive) or an exceedingly large number of variables $d$ (high-dimensional) or both.

The raw data $A$ is usually less interesting and informative than the spaces it defines, e.g. its row and column spaces or its principal subspaces. Moreoever, it often happens that $A$ can be well-approximated by a subspace $\mathbf{A} \in \operatorname{Gr}(k,n)$ where $k \ll m$ and $n \ll d$. The process of getting from $A$ to $\mathbf{A}$ is well-studied, e.g. randomly sample a subset of representative landmarks or compute principal components.

Subspace-valued data appears in a wide range of applications: computer vision [@MYDF; @VidMaSas2005], bioinformatics [@HH], machine learning [@HRS; @LZ], communication [@LHS; @ZT], coding theory [@AC; @BN; @CHS; @DHST], statistical classification [@HamLee2008], and system identification [@MYDF]. In computational mathematics, subspaces arise in the 