This section describes briefly the features supported by Lucid.

Legend

The color legend below describes the status of each feature in the library.

flowchart TB
    a["Available"]:::available
    i["In active development"]:::in-development
    p["Planned for the future"]:::planned
    n["Not planned"]:::not-planned

classDef available stroke:#0b0,stroke-width:2px;
classDef in-development stroke:#bb0,stroke-width:2px,stroke-dasharray:2px;
classDef planned stroke:#0bb,stroke-width:2px,stroke-dasharray:5px;
classDef not-planned stroke:#b00,stroke-width:2px,stroke-dasharray:15px;

Estimators

flowchart TB
    Estimator["Estimator"]
    u{{"Unsupervised"}}:::not-planned
    c{{"Classification"}}:::not-planned
    r{{"Regression"}}
    s{{"Supervised"}}
    p{{"Parametric"}}:::not-planned
    np{{"Non parametric"}}
    GP["Gaussian Process"]:::planned
    KR["Kernel Ridge"]:::available
    SVR["SVR"]:::planned

    Estimator --> s
    Estimator --> u
    s --> r
    s --> c
    r ---> np
    r --> p
    np --> KR
    np --> SVR
    np --> GP

classDef available stroke:#0b0,stroke-width:2px;
classDef in-development stroke:#bb0,stroke-width:2px,stroke-dasharray:2px;
classDef planned stroke:#0bb,stroke-width:2px,stroke-dasharray:5px;
classDef not-planned stroke:#b00,stroke-width:2px,stroke-dasharray:15px;

Definitions

• Unsupervised: Estimators that do not require labeled data for training.
• Supervised: Estimators that require labeled data for training.
• Classification: Estimators that predict discrete labels or categories.
• Regression: Estimators that predict continuous values.

Formulae

Kernel Ridge

A regression technique that combines ridge regression with kernel methods.

Loss Function

$$ \ell(w) = \frac{1}{2} |Y - w^T\Phi(X)|^2_2 + \frac{1}{2} \( \lambda \) n |w|^2_2 $$

where:

• \( Y \) is the target vector,
• \( X \) is the input data,
• \(\Phi(X)\) is the feature map over the input data,
• \( w \) is the weight vector we want to learn,
• \(\) \lambda \(\) is the regularization parameter,
• \( n \) is the number of samples.

To obtain the optimal weights, we solve the following equation:

$$ w = \Phi(X)(\Phi(X)^T\Phi(X) + \( \lambda \) n I)^{-1}Y $$

Prediction

To predict a new output \(\hat{y}\) for a new input \( x \), we use the following formula:

$$ \hat{y} = w^T \Phi(x) = y(\Phi(X)^T\Phi(X) + \( \lambda \) n I)^{-1}\Phi(x)^T\Phi(X) = y(K + \( \lambda \) n I)^{-1}\kappa(x) $$

where \( K \) is the Gram matrix computed from the training data, and \(\kappa(x)\) is the kernel function applied to the new input \( x \) against all training inputs.

Supported Kernels

flowchart TB
    gaussian["Gaussian"]:::available
    l["Linear"]:::planned
    p["Polynomial"]:::planned
    s["Sigmoid"]:::planned

classDef available stroke:#0b0,stroke-width:2px;
classDef in-development stroke:#bb0,stroke-width:2px,stroke-dasharray:2px;
classDef planned stroke:#0bb,stroke-width:2px,stroke-dasharray:5px;
classDef not-planned stroke:#b00,stroke-width:2px,stroke-dasharray:15px;