Vibration　monitoring　is　one　of　the　most　effective　ways　for　bearing　fault　diagnosis,　and　a　challenge　is　how　to　accurately　estimate　bearing　fault　signals　from　noisy　vibration　signals.　In　this　paper,　a　nonconvex　sparse　regularization　method　for　bearing　fault　diagnosis　is　proposed　based　on　the　generalized　minimax-concave　(GMC)　penalty,　which　maintains　the　convexity　of　the　sparsity-regularized　least　squares　cost　function,　and　thus　the　global　minimum　can　be　solved　by　convex　optimization　algorithms.　Furthermore,　we　introduce　a　k-sparsity　strategy　for　the　adaptive　selection　of　the　regularization　parameter.　The　main　advantage　over　conventional　filtering　methods　is　that　GMC　can　better　preserve　the　bearing　fault　signal　while　reducing　the　interference　of　noise　and　other　components;　thus,　it　can　significantly　improve　the　estimation　accuracy　of　the　bearing　fault　signal.　A　simulation　study　and　two　run-to-failure　experiments　verify　the　effectiveness　of　GMC　in　the　diagnosis　of　localized　faults　in　rolling　bearings,　and　the　comparison　studies　show　that　GMC　provides　more　accurate　estimation　results　than　L1-norm　regularization　and　spectral　kurtosis.