Cortical populations of neurons develop sparse representations adapted to the statistics of the environment. To learn efficient population codes, synaptic plasticity mechanisms must differentiate relevant latent features from spurious input correlations, which are omnipresent in cortical networks. Here, we develop a theory for sparse coding and synaptic plasticity that is invariant to second-order correlations in the input. Going beyond classical Hebbian learning, our learning objective explains the functional form of observed excitatory plasticity mechanisms, showing how Hebbian long-term depression (LTD) cancels the sensitivity to second-order correlations so that receptive fields become aligned with features hidden in higher-order statistics. Invariance to second-order correlations enhances the versatility of biologically realistic learning models, supporting optimal decoding from noisy inputs and sparse population coding from spatially correlated stimuli. In a spiking model with triplet spike-timing-dependent plasticity (STDP), we show that individual neurons can learn localized oriented receptive fields, circumventing the need for input preprocessing, such as whitening, or population-level lateral inhibition. The theory advances our understanding of local unsupervised learning in cortical circuits, offers new interpretations of the Bienenstock-Cooper-Munro and triplet STDP models, and assigns a specific functional role to synaptic LTD mechanisms in pyramidal neurons.