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Modern neural models trained on textual data rely on pre-trained representations that emerge without direct supervision. As these representations are increasingly being used in real-world applications, the inability to \emph{control} their content becomes an increasingly important problem. In this work, we formulate the problem of identifying a linear subspace that corresponds to a given concept, and removing it from the representation. We formulate this problem as a constrained, linear minimax game, and show that existing solutions are generally not optimal for this task. We derive a closed-form solution for certain objectives, and propose a convex relaxation that works well for others. When evaluated in the context of binary gender removal, the method recovers a low-dimensional subspace whose removal mitigates bias by intrinsic and extrinsic evaluation. Surprisingly, we show that the method---despite being linear---is highly expressive, effectively mitigating bias in the output layers of deep, nonlinear classifiers while maintaining tractability and interpretability.

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