Although the ability to reliably identify objects over a large range of stimulus intensities is a fundamental feature of all sensory systems, the neural circuit mechanisms that implement intensity invariance remain poorly understood. In mammals, odors are detected by individual olfactory sensory neurons expressing just one out of ~1,000 different types of odorant receptor, each of which projects to a specific pair of glomeruli in the olfactory bulb (OB). At low concentrations, odorants selectively bind high-affinity receptors, activating a sparse combination of responsive glomeruli. However, receptor activation, and thus glomerular activation, becomes less specific at higher odorant concentrations (1-4), potentially degrading the representation of odor identity. However, psychophysical studies indicate that odors retain their perceptual identities while concentration varies over several orders of magnitude (5-7). The olfactory system must therefore transform these concentration-dependent odor responses at early stages of processing into concentration-invariant representations of odor identity.Animals rely on olfaction to find food, attract mates and avoid predators. To support these behaviors, animals must reliably identify odors across different odorant concentrations. The neural circuit operations that implement this concentration invariance remain unclear. Here we demonstrate that, despite concentration-dependence in olfactory bulb (OB), representations of odor identity are preserved downstream, in piriform cortex (PCx). The OB cells responding earliest after inhalation drive robust responses in a sparse subset of PCx neurons. Recurrent collateral connections broadcast their activation across PCx, recruiting strong, global feedback inhibition that rapidly suppresses cortical activity for the remainder of the sniff, thereby discounting the impact of slower, concentration-dependent OB inputs. Eliminating recurrent collateral output dramatically amplifies PCx odor responses, renders cortex steeply concentration-dependent, and abolishes concentration-invariant identity decoding.