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Interference alignment (IA) is a transmission technique for exploiting all available degrees of freedom in the frequency- or time-selective interference channel with an arbitrary number of users. Most prior work on IA, however, neglects interference from other nodes in the network that are not participating in the alignment operation. This paper proposes three generalizations of IA for the multiple-antenna interference channel with multiple users that account for colored noise, which models uncoordinated interference. First, a minimum interference-plus-noise leakage (INL) algorithm is presented and shown to be equivalent to previous subspace methods when noise is spatially white or negligible. This algorithm results in orthonormal precoders that are desirable for practical implementation with limited feedback. A joint minimum mean square error design that jointly optimizes the transmit precoders and receive spatial filters is then proposed, whereas previous designs neglect the receive spatial filter. Finally, a maximum signal-to-interference-plus-noise ratio (SINR) algorithm is developed and proven to converge, unlike previous maximum SINR algorithms. The sum throughput of these algorithms is simulated in the context of a network with uncoordinated cochannel interferers that are not participating in the alignment protocol. It is found that a network with cochannel interference can benefit from employing precoders that are designed to consider that interference, but in extreme cases, such as when only one receiver has a large amount of interference, ignoring that the cochannel interference is advantageous.