# Learning Manifolds using Gradient Descent

I have a feedforward neural network $$F(W): \mathbb R^d \rightarrow \mathbb R^k$$ with $$Relu$$ activation, $$m$$ neurones per layer, $$L$$ layers and softmax on the output layer. $$W$$ denotes the weight matrices.

My training data are points concentrated in compact and connected manifolds $$M_i \in \mathbb R^d$$. Each manifold has diameter at most $$D$$, which means that if $$x_1, x_2 \in M$$, the distance between $$x_1$$ and $$x_2$$ is at most $$D$$. Let's consider $$D << 1$$.

When I run gradient descent to minimise the loss function $$L(W,x_1) =\frac 1 2 \| F(W,x_1) - y \|^2$$, I know that $$L(x_1)$$ decreases by $$\eta \nabla L(x_1)$$, where $$\eta$$ is the step size.

Assuming that $$x_1$$ and $$x_2$$ belong to the same manifold $$M$$, what can I say about $$L(W,x_2)$$? does it decrease when I run gradient descent on $$L(W,x_1)$$?

In other words, how does the change of weights given by one iteration of gradient descent on $$L(W,x_1)$$ affects the loss function computed on points close to $$x_1$$?

Is it reasonable to say that $$L(W,x_2)$$ is expected to decrease as well or not?