Low-rank subspace clustering (LRSC) has been considered as the state-of-the-art method on small datasets. LRSC constructs a desired similarity graph by low-rank representation (LRR), and employs a spectral clustering to segment the data samples. However, effectively applying LRSC into clustering big data becomes a challenge because both LRR and spectral clustering suffer from high computational cost. To address this challenge, we create a projective low-rank subspace clustering (PLrSC) scheme for large scale clustering problem. First, a small dataset is randomly sampled from big dataset. Second, our proposed predictive low-rank decomposition (PLD) is applied to train a deep encoder by using the small dataset, and the deep encoder is used to fast compute the low-rank representations of all data samples. Third, fast spectral clustering is employed to segment the representations. As a non-trivial contribution, we theoretically prove the deep encoder can universally approximate to the exact (or bounded) recovery of the row space. Experiments verify that our scheme outperforms the related methods on large scale datasets in a small amount of time. We achieve the state-of-art clustering accuracy by 95.8% on MNIST using scattering convolution features.