Linear Discriminant Analysis on a transformed space

Let $$S$$ be a finite subset of a $$\mathbb{R}^k$$ partitioned into $$N$$ subsets $$S_1, \ldots, S_N$$ and let $$n_j = |S_j|$$. The between-groups sum of squares of the partition is defined as $$bSS(S_1,\ldots, S_N) = \sum_{j=1}^{N} n_j ||\mathbb{E}[S_j] - \mathbb{E}[S]||^2$$ The within-group sum of squares of the each $$S_j$$ is defined as $$SS(S_j) = \sum_{u \in S_j} ||u - \mathbb{E}[S_j]||^2$$ The sum of squares of the set $$S$$ is defined as $$SS(S) = \sum_{v \in S} ||v - \mathbb{E}[S]||^2$$

It is, I think, well-known that $$SS(S) = bSS(S_1,\ldots, S_n) + \sum_{j=1}^{N} SS(S_j)$$

Both Linear Discriminant Analysis (LDA) and Partial Least Squares Discriminant Analysis (PLSDA) share a common initial part that aims at finiding a one-dimensional subspace $$U$$ of $$\mathbb{R}^k$$ such that the projection map $$\pi_U$$ of $$\mathbb{R}^k$$ onto $$U$$ maximizes the quantity

$$\frac{bSS(\pi_U(S_1),\ldots, \pi_U(S_n))}{\sum_{j=1}^{N} SS(\pi_U(S_j))}$$

The single vector $$u$$ that generates $$U$$ is understood as the direction of best separation for the sets $$S_1,\ldots, S_n$$ and represents a latent variable resulting from a linear combination of the original features (that are associated to the canonical basis of $$\mathbb{R}^k$$).

Suppose $$S_1,\ldots, S_n$$ are not linearly separable in $$\mathbb{R}^k$$ but there is a real vector space $$W$$ and a non-linear function $$f: \mathbb{R}^k \to W$$ (such as that computed by the hidden layers of a deep neural network) such that the images $$f(S_1),\ldots, f(S_n)$$ are linearly separable in $$W$$.

Suppose one performs LDA or PLSDA in $$W$$ and finds a one-dimensional subspace $$U \subseteq W$$ that represents the most important direction for the separation of $$f(S_1),\ldots, f(S_n)$$. The anti-image $$f^{-1}(U)$$ is not in general a subspace of $$V$$. Are there situations in which $$f^{-1}(U)$$ can be used to understand directions of best separation inside the original input space $$\mathbb{R}^k$$ ?