We present a method for learning a low-dimensional manifold for speech from clean speech samples in high-dimensional space. Using this manifold, we perform speech denoising by projecting noisy speech onto the manifold to remove non-speech components. This method of denoising classifies our algorithm as a signal subspace denoising method, where high-dimensional noisy data is projected onto the signal subspace to recover the signal of interest. We ran denoising experiments with different types of additive noise. The proposed method not only recovers the second formant more accurately, but also produces denoised speech with higher quality (as illustrated by PESQ scores) compared to other signal subspace denoising algorithms.