Deep Neural Networks (DNN) are computationally intensive to train. It consists of a large number of multidimensional dot products between many weights and input vectors. However, there can be significant similarities among input vectors. If one input vector is similar to another, its computations with the weights are similar to those of the other and, therefore, can be skipped by reusing the already-computed results. We propose a novel scheme, called MERCURY, to exploit input similarity during DNN training in a hardware accelerator. MERCURY uses Random Projection with Quantization (RPQ) to convert an input vector to a bit sequence, called Signature. A cache (MCACHE) stores signatures of recent input vectors along with the computed results. If the Signature of a new input vector matches that of an already existing vector in the MCACHE, the two vectors are found to have similarities. Therefore, the already-computed result is reused for the new vector. To the best of our knowledge, MERCURY is the first work that exploits input similarity using RPQ for accelerating DNN training in hardware. The paper presents a detailed design, workflow, and implementation of the MERCURY. Our experimental evaluation with twelve different deep learning models shows that MERCURY saves a significant number of computations and speeds up the model training by an average of 1.97× with an accuracy similar to the baseline system.