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A fundamental question in many data analysis settings is the problem ofdiscerning the "natural" dimension of a data set. That is, when a data set isdrawn from a manifold (possibly with noise), a meaningful aspect of the data isthe dimension of that manifold. Various approaches exist for estimating thisdimension, such as the method of Secant-Avoidance Projection (SAP).Intuitively, the SAP algorithm seeks to determine a projection which bestpreserves the lengths of all secants between points in a data set; by applyingthe algorithm to find the best projections to vector spaces of variousdimensions, one may infer the dimension of the manifold of origination. Thatis, one may learn the dimension at which it is possible to construct adiffeomorphic copy of the data in a lower-dimensional Euclidean space. UsingWhitney's embedding theorem, we can relate this information to the naturaldimension of the data. A drawback of the SAP algorithm is that a data set with$T$ points has $O(T^2)$ secants, making the computation and storage of allsecants infeasible for very large data sets. In this paper, we propose a novelalgorithm that generalizes the SAP algorithm with an emphasis on addressingthis issue. That is, we propose a hierarchical secant-baseddimensionality-reduction method, which can be employed for data sets whereexplicitly calculating all secants is not feasible.

Too many secants: a hierarchical approach to secant-based dimensionality reduction on large data sets