Date: 2025-11-26

# Kernel Logistic PLS

#### Frédéric Bertrand

#### 2025-11-26

## Kernel Logistic PLS (klogitpls)

We first extract latent scores with Kernel PLS (KPLS):

\[ T = K_c U, \]

where \(K_c = H K(X,X) H\) is the centered Gram matrix and the columns of \(U\) are the dual score directions (KPLS deflation).

We then fit a logistic link in the latent space using IRLS:

\[ \eta = \beta_0 + T \beta, \qquad p = \sigma(\eta), \] \[ W = \mathrm{diag}(p (1-p)), \qquad z = \eta + \frac{y - p}{p(1-p)}. \]

At each iteration, solve the weighted least squares system for \([\beta_0, \beta]\): \[ (\tilde{M}^\top \tilde{M}) \theta = \tilde{M}^\top \tilde{z}, \quad \tilde{M} = W^{1/2}[1, T], \ \tilde{z} = W^{1/2} z. \]

Optionally, we alternate: replace \(y\) by \(p\) and recompute KPLS to refresh \(T\) for a few steps.
Prediction on new data uses the centered cross-kernel \(K_c(X_\*, X)\) and the stored KPLS basis \(U\): \[ T_\* = K_c(X_\*, X) \, U, \qquad \hat{p}_\* = \sigma\!\big(\beta_0 + T_\* \beta\big). \]