Kernel Width

Kernel width determines how quickly sample weights decay with cosine distance or Euclidean distance from the original instance. A small kernel width creates a tight, highly local neighborhood where only very similar perturbed samples receive significant weight. A large kernel width flattens the weighting function, allowing more distant samples to influence the surrogate model training. This parameter effectively defines what 'local' means for a specific explanation, controlling the scope of the local decision boundary being approximated.

Kernel width directly mediates the fidelity-interpretability trade-off in local explanations:

• Narrow kernel: Captures highly local, non-linear decision boundaries with greater precision but produces unstable explanations sensitive to sampling noise
• Wide kernel: Produces smoother, more stable explanations that approximate global behavior but may miss critical local variations
• Optimal selection: Requires balancing the complexity of the underlying model's decision surface against the need for reproducible explanations The choice fundamentally determines whether the explanation captures genuine local structure or averages over a broader region.

LIME employs an exponential kernel to convert distances into sample weights: weight = exp(-D(x, z)² / σ²), where σ (sigma) is the kernel width and D(x, z) is the distance between the original instance and a perturbed sample. This function ensures:

• Samples identical to the original receive maximum weight of 1.0
• Weights decay smoothly and exponentially with increasing distance
• The rate of decay is controlled entirely by the kernel width parameter
• Very distant samples receive near-zero weight, effectively excluding them from surrogate model training

Kernel width is the primary control for explanation stability across multiple runs with different random seeds. A kernel that is too narrow causes the explanation to overfit to the specific random perturbation samples generated in each run, producing inconsistent feature importance rankings. A kernel that is too wide produces stable but potentially misleading explanations that fail to capture local nuances. OptiLIME addresses this by automatically selecting the kernel width that maximizes stability while maintaining a minimum fidelity threshold, ensuring reproducible explanations.

Optimal kernel width varies significantly by data modality and feature representation:

• Text data: Typically uses larger kernel widths due to the high-dimensional, sparse nature of TF-IDF or embedding representations where cosine distances can be large
• Image data: Requires careful tuning against superpixel segmentation granularity, as the distance between superpixel masks affects neighborhood density
• Tabular data: Depends on feature scaling and the density of the perturbation sampling strategy around the instance
• Categorical features: May require specialized distance metrics that account for non-numeric feature spaces

Bayesian LIME extends kernel width selection by incorporating uncertainty quantification. Rather than using a fixed kernel width, Bayesian approaches treat it as a random variable with a prior distribution, allowing the explanation to express confidence intervals around feature importance scores. This integration reveals when the chosen kernel width may be inappropriate—wide confidence intervals suggest the explanation is sensitive to the kernel width choice, indicating potential instability in the local approximation that practitioners should investigate before trusting the explanation.

The weighting function that operationalizes kernel width. It assigns a proximity weight to each perturbed sample using the formula exp(-D(x, z)^2 / sigma^2), where sigma is the kernel width. Samples closer to the original instance receive weights approaching 1.0, while distant samples decay exponentially toward zero. This enforces the locality constraint, ensuring the surrogate model prioritizes fitting the immediate neighborhood.

The proximity metric used in LIME for text data. It measures the angle between TF-IDF vector representations of the original document and perturbed samples, producing a value between 0 (identical) and 1 (orthogonal). Unlike Euclidean distance, cosine distance ignores document length differences, making it robust for comparing texts of varying sizes after token masking perturbations.

The property that feature importance rankings remain consistent across multiple runs with different random seeds. An excessively narrow kernel width produces unstable explanations because the surrogate model overfits to a tiny, noisy neighborhood. Widening the kernel increases stability by incorporating more samples, but risks violating the local fidelity assumption if the decision boundary is highly non-linear.

The statistical backbone of LIME. This non-parametric method fits a simple model to a localized subset of data, weighting each training point by its proximity to the query instance. The kernel width directly controls the bandwidth of this regression: a small bandwidth captures fine-grained local structure but suffers high variance, while a large bandwidth produces smoother, lower-variance estimates that may miss sharp decision boundaries.

The fundamental balancing act that kernel width mediates. A narrow kernel yields high local fidelity because the surrogate only needs to be accurate in a tiny region, but the explanation may be unstable and fail to capture broader patterns. A wide kernel produces more globally representative explanations but may smooth over critical local decision boundary details. The optimal width is instance-dependent and reflects the local curvature of the model's prediction function.