1,241 research outputs found
A Comparative Study of Efficient Initialization Methods for the K-Means Clustering Algorithm
K-means is undoubtedly the most widely used partitional clustering algorithm.
Unfortunately, due to its gradient descent nature, this algorithm is highly
sensitive to the initial placement of the cluster centers. Numerous
initialization methods have been proposed to address this problem. In this
paper, we first present an overview of these methods with an emphasis on their
computational efficiency. We then compare eight commonly used linear time
complexity initialization methods on a large and diverse collection of data
sets using various performance criteria. Finally, we analyze the experimental
results using non-parametric statistical tests and provide recommendations for
practitioners. We demonstrate that popular initialization methods often perform
poorly and that there are in fact strong alternatives to these methods.Comment: 17 pages, 1 figure, 7 table
Collective atomic scattering and motional effects in a dense coherent medium
We investigate collective emission from coherently driven ultracold 88Sr atoms. We perform two sets of experiments using a strong and weak transition that are insensitive and sensitive, respectively, to atomic motion at 1 μK. We observe highly directional forward emission with a peak intensity that is enhanced, for the strong transition, by >103 compared with that in the transverse direction. This is accompanied by substantial broadening of spectral lines. For the weak transition, the forward enhancement is substantially reduced due to motion. Meanwhile, a density-dependent frequency shift of the weak transition (∼10% of the natural linewidth) is observed. In contrast, this shift is suppressed to <1% of the natural linewidth for the strong transition. Along the transverse direction, we observe strong polarization dependences of the fluorescence intensity and line broadening for both transitions. The measurements are reproduced with a theoretical model treating the atoms as coherent, interacting radiating dipoles
Fraenkel's Partition and Brown's Decomposition
Denote the sequence ([ (n-x') / x ])_{n=1}^\infty by B(x, x'), a so-called
Beatty sequence. Fraenkel's Partition Theorem gives necessary and sufficient
conditions for B(x, x') and B(y, y') to tile the positive integers, i.e., for
B(x, x') \cap B(y, y') = \emptyset and B(x, x') \cup B(y, y') = {1,2, 3, ...}.
Fix 0 < x < 1, and let c_k = 1 if k \in B(x, 0), and c_k = 0 otherwise, i.e.,
c_k=[ (k+1) / x ] - [ k / x]. For a positive integer m let C_m be the binary
word c_1c_2c_3... c_m. Brown's Decomposition gives integers q_1, q_2, ...,
independent of m and growing at least exponentially, and integers t, z_0, z_1,
z_2, ..., z_t (depending on m) such that C_m =
C_{q_t}^{z_t}C_{q_{t-1}}^{z_{t-1}} ... C_{q_1}^{z_1}C_{q_0}^{z_0}. In other
words, Brown's Decomposition gives a sparse set of initial segments of C_\infty
and an explicit decomposition of C_m (for every m) into a product of these
initial segments.Comment: 19 page
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