Complex linear projection for acoustic modeling

A system comprising:

• data processing hardware;
• and
• memory hardware in communication with the data processing hardware, the memory hardware storing instructions that when executed on the data processing hardware cause the data processing hardware to perform operations comprising:
• receiving a training data set comprising audio data corresponding to a plurality of different training utterances;
• and
• for each training utterance in the training data set:
• converting the corresponding audio data into frequency domain data;
• processing, using a frequency domain filter, the frequency domain data to generate a frequency feature representation of the training utterance;
• generating, using a speech recognition model based on the frequency feature representation, a probability distribution over possible recognition outputs for the training utterance;
• and
• updating parameters of the frequency domain filter and parameters of the speech recognition model jointly based on the probability distribution over possible recognition outputs for the training utterance and a labeled recognition output for the training utterance.