Effective　fault　detection　and　diagnosis　(FDD)　is　crucial　in　modern　industry　applications,　but　developing　accurate　and　interpretable　FDD　models　for　complex　processes　remains　a　challenge.　In　this　article,　we　propose　a　novel　approach　that　partitions　measurements　into　a　principal　component　and　a　residual　subspace,　formulating　the　FDD　model-building　problem　as　a　separable　nonlinear　optimization　problem.　To　improve　model　interpretability,　we　incorporate　sparse　regularizers　to　derive　sparse　loadings.　Taking　advantage　of　the　special　separable　structure　presented　in　such　problems,　we　present　an　efficient　variable　projection-based　algorithm　that　significantly　reduces　the　dimension　of　the　parameters　by　solving　a　least-squares　problem　with　an　orthonormal　constraint.　This　results　in　a　reduced　problem　that　only　contains　loading　parameters.　We　further　establish　a　significant　theorem,　which　is　essential　for　computing　the　gradient　of　the　reduced　objective　function　and　addressing　the　coupling　between　different　parts　of　the　parameters,　leading　to　improved　identification　performance　of　the　proposed　algorithm.　Numerical　results　on　synthetic　and　real-world　datasets　demonstrate　that　our　algorithm　achieves　faster　convergence　speed　and　better　evaluation　metrics.