1,458,378 research outputs found
The Complexity of the k-means Method
The k-means method is a widely used technique for clustering points in Euclidean space. While it is extremely fast in practice, its worst-case running time is exponential in the number of data points. We prove that the k-means method can implicitly solve PSPACE-complete problems, providing a complexity-theoretic explanation for its worst-case running time. Our result parallels recent work on the complexity of the simplex method for linear programming
Randomized Dimensionality Reduction for k-means Clustering
We study the topic of dimensionality reduction for -means clustering.
Dimensionality reduction encompasses the union of two approaches: \emph{feature
selection} and \emph{feature extraction}. A feature selection based algorithm
for -means clustering selects a small subset of the input features and then
applies -means clustering on the selected features. A feature extraction
based algorithm for -means clustering constructs a small set of new
artificial features and then applies -means clustering on the constructed
features. Despite the significance of -means clustering as well as the
wealth of heuristic methods addressing it, provably accurate feature selection
methods for -means clustering are not known. On the other hand, two provably
accurate feature extraction methods for -means clustering are known in the
literature; one is based on random projections and the other is based on the
singular value decomposition (SVD).
This paper makes further progress towards a better understanding of
dimensionality reduction for -means clustering. Namely, we present the first
provably accurate feature selection method for -means clustering and, in
addition, we present two feature extraction methods. The first feature
extraction method is based on random projections and it improves upon the
existing results in terms of time complexity and number of features needed to
be extracted. The second feature extraction method is based on fast approximate
SVD factorizations and it also improves upon the existing results in terms of
time complexity. The proposed algorithms are randomized and provide
constant-factor approximation guarantees with respect to the optimal -means
objective value.Comment: IEEE Transactions on Information Theory, to appea
Skill set profile clustering: the empty K-means algorithm with automatic specification of starting cluster centers
While studentsā skill set profiles can be estimated with formal cognitive diagnosis models [8], their computational complexity makes simpler proxy skill estimates attractive [1, 4, 6]. These estimates can be clustered to generate groups of similar students. Often hierarchical agglomerative clustering or k-means clustering is utilized, requiring, for K skills, the specification of 2^K clusters. The number of skill set profiles/clusters can quickly become computationally intractable. Moreover, not all profiles may be present in the population. We present a flexible version of k-means that allows for empty clusters. We also specify a method to determine efficient starting centers based on the Q-matrix. Combining the two substantially improves the clustering results and allows for analysis of data sets previously thought impossible
A strong direct product theorem for quantum query complexity
We show that quantum query complexity satisfies a strong direct product
theorem. This means that computing copies of a function with less than
times the quantum queries needed to compute one copy of the function implies
that the overall success probability will be exponentially small in . For a
boolean function we also show an XOR lemma---computing the parity of
copies of with less than times the queries needed for one copy implies
that the advantage over random guessing will be exponentially small.
We do this by showing that the multiplicative adversary method, which
inherently satisfies a strong direct product theorem, is always at least as
large as the additive adversary method, which is known to characterize quantum
query complexity.Comment: V2: 19 pages (various additions and improvements, in particular:
improved parameters in the main theorems due to a finer analysis of the
output condition, and addition of an XOR lemma and a threshold direct product
theorem in the boolean case). V3: 19 pages (added grant information
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