Anti-crosstalk

The full non-linear derivation

For warm up we linearize in the simplest possible way. Suppose we allow only $\mathbf m_1$ to vary keeping $\mathbf m_2$ fixed. We put $\mathbf d_2$ on the diagonal of a matrix, say $\mathbf D_2$. The regression for anti-crosstalk is now

$$\mathbf 0 \approx \mathbf T \mathbf D_2\mathbf d_1$$ (6)

$$\mathbf 0 \approx \mathbf T \mathbf D_2\mathbf F_1 \mathbf m_1$$ (7)

Define the element-by-element cross product of $\mathbf d_2$ times $\mathbf d_1$ to be $\mathbf d_1 \times \mathbf d_2$. Now let us linearize the full non-linear anti-crosstalk regularization. Let a single element of $\mathbf d_1 \times \mathbf d_2$ be decomposed as a base plus a perturbation $d = \bar d +\tilde d$. A single component of the vector $\mathbf d_1 \times \mathbf d_2$ is $(\bar d_1+\tilde d_1) (\bar d_2+\tilde d_2)$. Linearizing the product (neglecting the product of the perturbations) gives

$$\bar d_2 \tilde d_1 + \bar d_1 \tilde d_2 + \bar d_2 \bar d_1$$ (8)

This is one component. We seek an expression for all. It will be a vector which is a product of a matrix with a vector. We want no unknowns in matrices; we want them all in vectors so we will know how to solve for them.

$$\bar{\mathbf{D}}_2 \tilde{\mathbf{d}}_1 + \bar{\mathbf{D}}_1 \tilde{\mathbf{d}}_2 + \bar{\mathbf{D}}_1 \bar{\mathbf{d}}_2$$ (9)

Express the perturbation parts of the vectors as functions of the model space

$$\bar{\mathbf {D}}_2 \mathbf{ F}_1 \tilde{{\mathbf m}}_1 + \bar{\m... ...1 \mathbf{ F}_2 \tilde{{\mathbf m}}_2+ \bar{\mathbf {D}}_1 \bar{ {\mathbf d}}_2$$ (10)

This vector should be viewed under many windows (triangle shaped, for example). Under each window we hope to see the product have a small value. The desired anti-crosstalk regression is to minimize the length of the vector below by variation of the model parameters $\tilde{\mathbf m}_1$ and $\tilde{\mathbf m}_2$.

$$\mathbf 0 \quad\approx\quad \mathbf T( \bar{\mathbf D}_2 \mathbf ... ...hbf D}_1 \mathbf F_2 \tilde{\mathbf m}_2+ \bar{\mathbf D}_1 \bar{ \mathbf d}_2)$$ (11)

This regression augments our usual regularizations. Perhaps it partially or significantly supplants them. Unfortunately, it requires yet another epsilon.

Upon finding $\tilde{\mathbf m}_1$ and $\tilde{\mathbf m}_2$ we update the base model $\bar{\mathbf m} \leftarrow \bar{\mathbf m} + \tilde{\mathbf m}$ and iterate.

Anti-crosstalk