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The computational cost for most classification algorithms is dependent on the dimensionality of the input samples. As the dimensionality could be high in many cases, particularly those associated with image classification, reducing the dimensionality of the data becomes a necessity. The traditional dimensionality reduction methods are data dependent, which poses certain practical problems. Random projection (RP) is an alternative dimensionality reduction method that is data independent and bypasses these problems. The nearest neighbor classifier has been used with the RP method in classification problems. To obtain higher recognition accuracy, this study looks at the robustness of RP dimensionality reduction for several recently proposed classifiers - sparse classifier (SC), group SC (along with their fast versions), and the nearest subspace classifier. Theoretical proofs are offered regarding the robustness of these classifiers to RP. The theoretical results are confirmed by experimental evaluations.