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We consider the problem of partitioning a set of m points in the n-dimensional Euclidean space into k clusters (usually m and n are variable, while k is fixed), so as to minimize the sum of squared distances between each point and its cluster center. This formulation is usually called k-means clustering (KMN + 00). We prove that this problem in NP-hard even(More)
We investigate the phenomenon of depth-reduction in commutative and non-commutative arithmetic circuits. We prove that in the commutative setting, uniform semi-unbounded arithmetic circuits of logarithmic depth are as powerful as uniform arithmetic circuits of polynomial degree (and unrestricted depth); earlier proofs did not work in the uniform setting.(More)
We extend the lower bound techniques of [17], to the unbounded-error probabilistic model. A key step in the argument is a generalization of Nepom-njašči˘ ı's theorem from the Boolean setting to the arithmetic setting. This generalization is made possible, due to the recent discovery of logspace-uniform TC 0 circuits for iterated multiplication [11]. Here is(More)
We consider the problem of dividing a set of m points in Euclidean n?space into k clusters (m; n are variable while k is xed), so as to minimize the sum of distances squared of each point to its \cluster center". This formulation diiers in two ways from the most frequently considered clustering problems in the literature, namely, here we have k xed and m; n(More)