---
id: "implement-lfm-kernel-core"
title: "Implement core LFM kernel functionality"
status: "Proposed"
priority: "High"
created: "2025-08-15"
last_updated: "2025-08-15"
owner: "Neil Lawrence"
github_issue: ""
dependencies: "design-modern-lfm-kernel"
tags:
- lfm
- kernel
- implementation
- core
---

# Implement core LFM kernel functionality

## Description
Implement the core LFM kernel class with basic functionality including kernel computation, parameter handling, and gradient computation.

## Background
- Design phase completed with modern LFM kernel architecture
- Need to implement the core kernel computation methods
- Should follow the mathematical foundations from the papers and MATLAB implementation

## Implementation Tasks
- [ ] Create `GPy.kern.LFM` class inheriting from appropriate base class
- [ ] Implement parameter handling for mass, damper, spring, sensitivity, delay
- [ ] Implement `K()` method for kernel matrix computation
- [ ] Implement `Kdiag()` method for diagonal computation
- [ ] Add parameter constraints and transformations
- [ ] Implement basic gradient computation
- [ ] Add support for different base kernels for latent functions

## Core Methods to Implement
- [ ] `__init__()` - Parameter initialization and validation
- [ ] `K(X, X2=None)` - Kernel matrix computation
- [ ] `Kdiag(X)` - Diagonal computation
- [ ] `update_gradients_full()` - Gradient computation
- [ ] `update_gradients_diag()` - Diagonal gradient computation
- [ ] `parameters_changed()` - Parameter update handling

## Acceptance Criteria
- [ ] Core LFM kernel class implemented and functional
- [ ] Basic kernel computation working correctly
- [ ] Parameter handling and constraints implemented
- [ ] Gradient computation implemented
- [ ] Unit tests passing for core functionality
- [ ] Integration with GPy's parameterization system

## Implementation Notes
- Follow the mathematical structure from the MATLAB implementation
- Use GPy's parameterization system for constraints
- Implement efficient computation methods
- Ensure proper handling of edge cases and numerical stability
- Add comprehensive docstrings and documentation

## Related
- CIP: 0001 (LFM kernel implementation)
- Backlog: design-modern-lfm-kernel
- Papers: Álvarez et al. (2009, 2012)