Suppose we have an $n$-dimenosional vector $\vec{x}\in\mathbb{R}^{n}$ representing an image. Furthermore, assume there is a function $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$ describing some sort of transformation on $\vec{x}$ (e.g. affine transformation), where $m\gg n$. If we assume that the vector $\vec{x}$ encodes some objects like for instance cars, how would we econde those prior information about objects (i.e. geometry, shapes, sizes) into the higher-dimensional representation of $\vec{x}$? Is there a way to preserve or encode information such as shapes and geometry about objects in higher-dimensions (topological spaces)?
To be more precise think of $f$ representing a composition of parametric differentiable non-linear functions such that $f:= f_{n}(\theta_{n}),\circ\ldots\circ,f_{1}(\theta_{1})$. For the sake of argument suppose that $f$ is a neural network which embeds $\vec{x}$ in a manifold. To my understanding (please correct me if I'm wrong) regardless of $f$ data such as $\vec{x}$ are encoded as points in a manifold (probably equiped with some properties) or higher dimenisonal spasce? Hence, the question is it possible to encode additional information about what $\vec{x}$ represents?
Thanks!
The answer depends to a large extent on the properties of the function $f$. The Euclidean space has several possible frameworks of structure: linear, topological, metric, differentiable-manifold, and algebraic, as well as fruitful combination of some of these. Information encoded in an $n$-dimensional vector could be different from information encoded in another $n$-dimensional vector, for several reasons: it could be "far" in a metric sense, or it could be nonequivalent with respect to a certain algebraic or topological equivalence relation. Thus, in order to provide a reasonable answer to your question, you should provide some information about $f$, in terms of how does it correspond to any of the above structures: is it linear? is it continuous? is differentiable? is it a group-homomorphism? etc.