This note isolates my own formalization from the broader research context in which I worked with Marzieh Eidi and Jürgen Jost. The model treats dimension reduction as a composition of a nonlinear deformation, a linear projection, and a change of coordinates. Alongside that factorization, I formulated a data-driven local approximation of a dimension-reduction rule using first- and second-order terms, then interpolated those local models across nearby inputs. The focus here is this decomposition and its mathematical structure, not a summary of the wider collaboration.
1. Treat dimension reduction as a map-valued rule
Let \(n>k\). Rather than regard dimension reduction only as a single embedding of one point cloud, model the procedure as a rule that takes an input distribution or data measure \(\mu\) to a map \(f:\mathbb R^n\to\mathbb R^k\):
This perspective makes the dependence of the dimension-reduction map on the input data part of the object being studied. The goal is not merely to describe one output embedding, but to build a model of how the rule changes as its input changes.
2. A finite-dimensional surrogate
To make the problem tractable, restrict the inputs to a finite-dimensional space. Choose \(M\) cubes from a uniform partition of \([0,1]^n\), and let \(\mu_j\) be the uniform measure supported on the \(j\)-th cube. Set
On the output side, choose continuous, linearly independent feature maps \(\phi_i:\mathbb R^n\to\mathbb R^k\), \(i=1,\ldots,N\), and let
The restricted rule is then a map between finite-dimensional spaces, \(F:X\to Y\). An input is represented by coefficients in the measure basis, and an output map by coefficients in the feature basis. This turns a map-valued learning problem into a finite-dimensional approximation problem.
3. Local second-order model
Fix an anchor \(x_0\in X\). For inputs near \(x_0\), approximate the rule by a second-order expansion:
where \(J:X\to Y\) is linear and \(H:X\times X\to Y\) is a symmetric bilinear map. In coordinates, if \(a\) describes the input perturbation and \(b\) the output coefficients, the model has the form
Given evaluations \(y_s=F(x_s)\) at perturbations around the anchor, the coefficients of \(J\) and \(H\) can be estimated by solving a linear system. The quality of this estimate depends on the perturbations being sufficiently informative and the system being well-conditioned; the second-order truncation also imposes an approximation error.
4. Interpolating local models
A single Taylor model is local. To approximate the rule near a query \(x\), select nearby anchors \(x_1,\ldots,x_\ell\) and nonnegative weights satisfying \(\sum_i\alpha_i=1\). Approximate the output by
with each anchor output itself estimated using its local linear and quadratic terms. This combines local information without assuming that one expansion is accurate throughout the input space. The choice of anchors and interpolation weights becomes part of the approximation design.
5. Factorization into deformation, projection, and coordinates
For a fixed input, let \(f:\mathbb R^n\to\mathbb R^k\) be the resulting dimension-reduction map. The proposed structural model is.
where
- \(N:\mathbb R^n\to\mathbb R^n\) is a nonlinear preprocessing map;
- \(P:\mathbb R^n\to H\) is the orthogonal projection onto a \(k\)-dimensional subspace \(H\subseteq\mathbb R^n\);
- \(T:H\to\mathbb R^k\) is a linear isomorphism that expresses the projected coordinates in the output space.
The decomposition separates three roles that are often bundled together in a dimension-reduction algorithm. The nonlinear stage can reorganize or flatten the geometry; projection removes dimensions; and the final isomorphism changes coordinates without losing information within the projected subspace.
6. Modelling the nonlinear stage as a flow
The preprocessing map \(N\) can be viewed as the time-one map of a flow. A gradient-flow form is
where the energy \(E\) is chosen to drive the initial point toward the desired transformed representation. This gives a language for describing the nonlinear stage as a continuous deformation before projection, rather than as an opaque preprocessing operation.
7. What this formalization does—and does not—claim
The framework combines two ideas: a finite-dimensional, data-driven surrogate for how the reduction rule varies with its input, and a structural factorization of a fixed reduction map into nonlinear deformation, projection, and coordinate change. These ideas provide a common language for discussing local approximation and geometric preprocessing.
The formalization is a modelling framework, not by itself a convergence theorem or a guarantee that a chosen flow preserves topology. To turn it into a complete algorithm, one must specify the feature basis, the anchor-selection and interpolation rules, the energy functional, and the conditions under which the estimated local models are stable. These are substantive mathematical and computational choices, not details that can be omitted.
The central idea is the decomposition itself: study the nonlinear geometric transformation separately from the unavoidable dimensional projection, while using local second-order models to approximate how the overall reduction rule responds to changes in its input.