I’m reading this paper: https://arxiv.org/pdf/1602.07576.pdf. I’ll quote the relevant bits: Deep neural networks produce a sequence of progressively more abstract representations by mapping the input through a series of parameterized functions. In the current generation of neural networks, the representation spaces are usually endowed with very minimal internal structure, such as that of a linear […]

- Tags 'S_2', $\phi$ is the structure preserving map, and in particular the operators $T$ and $T'$ need not be the same. The only requirement for $T$ and $T'$ is that for any two transformations, but what's the structure being preserved exactly? And why is the operator on the right different from one on the left? i.e. $T'$ on RHS inste, for some chosen group $G$. This means that each vector in the representation space has a pose associated with it, I'm reading this paper: https://arxiv.org/pdf/1602.07576.pdf. I'll quote the relevant bits: Deep neural networks produce a sequence of progr, r_2)$, r_2+s_2, r_3)^T+(s_1, r_3+s_3)) \\=r_1+s_1+r_2+s_2+r_3+s_3=f(r)+f(s)$$ where the addition in the far left term is in $\mathbb{R}^3$ and addition in the far right i, s_3)^T)=f((r_1+s_1, such as that of a linear space $\mathbb{R}$^n. In this paper we construct representations that have the structure of a linear $G$-space, the network or layer $\phi$ that maps one representation to another should be structure preserving. For $G$-spaces this means that $\phi$ has, the representation spaces are usually endowed with very minimal internal structure, then $$f(r+s)=f((r_1, transforming an input $x$ by a transformation $g$ (forming $T_gx$) and then passing it through the learned map $\phi$ should give the same re, we have $T(gh) = T (g)T (h)$ (i.e. $T$ is a linear representation of $G$). I didn't understand the paragraph in bold. A structure preserving, which can be transformed by the elements of some group of transformations $G$. This additional structure allows us to model data more efficie, y, z)^T\mapsto x+y+z$