Question about definition 3.2

[imxtx]

Thank you for your great work. I have a question about definition 3.2:

To my understanding, the transport map $T_{m,l}$ maps C(source distribution) to D(target distribution). Why do we map C(source distribution) to a new C(source distribution) and D(target distribution) to a new D(target distribution) using the map between the source and the target? Does it make any sense to compute the transport map in the next layer?

I find that there is a reverse mapping parameter in the LinearProj class, but I didn't find the code that calls it:

class LinearProj(nn.Module):
    def __init__(
        self,
        dim: int,
    ):
        super().__init__()
        self.w1 = nn.Parameter(torch.randn(1, dim))
        self.b1 = nn.Parameter(torch.zeros((1, dim)))

    def forward(self, x: torch.Tensor, reverse: bool = False):
        assert x.shape[-1] == self.w1.shape[-1]
        if not reverse:
            return x * self.w1 + self.b1
        else:
            return (x - self.b1) / (self.w1 + 1e-10)

Is this reverse mapping related to definition 3.2? Could you please elaborate on it? Thank you!

[prlz77]

Hello!

1. We estimate the transport for one layer ($l$)
2. We obtain transported outputs for $l$.
3. We estimate the transport on $l+1$ given the transported outputs from $l$.
4. This is done layer by layer in an incremental fashion until all transports have been estimated.

The reverse here is just a tool to help us map from target to source in case we'd like to. Imagine you learn to go from informal language to formal language. Now you can reverse and go from formal to informal.

Hope this helps.

[imxtx]

Hello!

1. We estimate the transport for one layer (
   l
   )
2. We obtain transported outputs for
   l
   .
3. We estimate the transport on
   l
   +
   1
   given the transported outputs from
   l
   .
4. This is done layer by layer in an incremental fashion until all transports have been estimated.

The reverse here is just a tool to help us map from target to source in case we'd like to. Imagine you learn to go from informal language to formal language. Now you can reverse and go from formal to informal.

Hope this helps.

Many thanks for your reply.

However, I'm still confused about the second step. Say we have C and D, the map $T_{C\rightarrow D}$ can transport C to D. In the following formulas, why do we use $T_{C\rightarrow D} (D)$ to get the transported outputs of D? Does it make sense to use the linear transportation from C to D to transport D and what do we get? Should we instead use $T_{D\rightarrow C}$ to transport D back to C?