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We present a new classification algorithm, principal component space analysis (PCNSA), which is designed for classification problems like object recognition where different classes have unequal and nonwhite noise covariance matrices. PCNSA first obtains a principal components subspace (PCA space) for the entire data. In this PCA space, it finds for each class "i", an Mi-dimensional subspace along which the class' intraclass variance is the smallest. We call this subspace an approximate space (ANS) since the lowest variance is usually "much smaller" than the highest. A query is classified into class "i" if its distance from the class' mean in the class' ANS is a minimum. We derive upper bounds on classification error probability of PCNSA and use these expressions to compare classification performance of PCNSA with that of subspace linear discriminant analysis (SLDA). We propose a practical modification of PCNSA called progressive-PCNSA that also detects "new" (untrained classes). Finally, we provide an experimental comparison of PCNSA and progressive PCNSA with SLDA and PCA and also with other classification algorithms-linear SVMs, kernel PCA, kernel discriminant analysis, and kernel SLDA, for object recognition and face recognition under large pose/expression variation. We also show applications of PCNSA to two classification problems in video-an action retrieval problem and abnormal activity detection.