Add LFM kernel implementation backlog items

- Add code review task for existing LFM implementations
- Add design task for modern LFM kernel architecture
- Add implementation task for core LFM kernel functionality
- Establish clear task dependencies and acceptance criteria
- Link to CIP-0001 and relevant papers

backlog/features/2025-08-15_design-modern-lfm-kernel.md

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+---
+id: "design-modern-lfm-kernel"
+title: "Design modern LFM kernel architecture"
+status: "Proposed"
+priority: "High"
+created: "2025-08-15"
+last_updated: "2025-08-15"
+owner: "Neil Lawrence"
+dependencies: ["lfm-kernel-code-review"]
+tags:
+- lfm
+- kernel
+- design
+- architecture
+---
+
+# Design modern LFM kernel architecture
+
+## Description
+Design a modern LFM kernel implementation that follows GPy's current architectural patterns and uses the multioutput kernel approach with output index as input.
+
+## Background
+- Current GPy LFM implementations don't use the modern multioutput kernel approach
+- Need to design a unified LFM kernel that integrates well with GPy's current framework
+- Should maintain backward compatibility while providing improved functionality
+
+## Design Requirements
+- [ ] Use GPy's multioutput kernel approach with output index as input
+- [ ] Follow consistent API design with other GPy kernels
+- [ ] Implement proper parameter handling and constraints
+- [ ] Support different base kernels for latent functions
+- [ ] Enable efficient gradient computation
+- [ ] Maintain backward compatibility with existing implementations
+
+## Design Tasks
+- [ ] Define kernel class structure and inheritance hierarchy
+- [ ] Design parameter handling for mass, damper, spring, sensitivity, delay
+- [ ] Plan integration with GPy's multioutput framework
+- [ ] Design cross-kernel computation methods
+- [ ] Plan parameter tying and constraint handling
+- [ ] Design efficient computation methods for large datasets
+
+## Acceptance Criteria
+- [ ] Complete design specification document
+- [ ] API design that follows GPy patterns
+- [ ] Integration plan with existing GPy infrastructure
+- [ ] Performance considerations documented
+- [ ] Backward compatibility strategy defined
+
+## Implementation Notes
+- Study how other multioutput kernels in GPy handle output indices
+- Consider parameter tying approaches from MATLAB implementation
+- Design for extensibility to different differential equation types
+- Plan for efficient computation of cross-kernel terms
+
+## Related
+- CIP: 0001 (LFM kernel implementation)
+- Backlog: lfm-kernel-code-review