Rotating　machinery　is　widely　used　in　industry.　However,　it　works　in　tough　environment,　which　makes　the　fault　features　extraction　difficult.　In　the　last　few　years,　sparse　representation,　as　a　kind　of　effective　feature　extraction　method,　has　great　promise　in　industrial　diagnosis.　As　for　the　traditional　sparse　representation,　the　greedy　algorithm　used　to　update　the　sparse　coefficients　is　prone　to　produce　local　optimal　solution　and　lead　to　over-fitting.　In　addition,　if　the　signal　contains　a　lot　of　redundant　information,　the　basis　learned　by　the　traditional　method　cannot　well　represent　the　fault　signal.　Aim　at　the　above　questions,　a　novel　weak　fault　diagnosis　method　based　on　sparse　representation　and　empirical　wavelet　transform　(EWT)　is　proposed　in　this　paper.　Gaussian　filter　is　exploited　to　process　the　signal　spectral,　which　can　make　the　signal　spectral　smooth　and　the　spectrum　division　more　precise.　Next,　the　signal　spectrum　is　divided　into　N　parts　based　on　EWT.　Then　the　kurtosis　is　utilized　to　screen　out　the　optimal　part　of　spectrum,　which　will　be　exploited　to　obtain　a　sparse　basis.　The　constraint　with　nuclear　norm　is　applied　to　remove　the　redundant　component　of　the　basis.　Finally,　the　LASSO　with　elastic　net,　is　employed　to　get　sparse　signal,　the　envelope　spectrum　is　used　to　extract　fault　feature.　Experimental　results　show　that　this　method　is　better　than　traditional　sparse　representation　using　learning　dictionary.　©　2020　IEEE.