The Fuzzy Kaehler Coset Space with the Darboux Coordinates

The Fedosov deformation quantization of the symplectic manifold is determined by a 1-form differential r. We identify a class of r for which the $\star$ product becomes the Moyal product by taking appropriate Darboux coordinates, but invariant by canonically transforming the coordinates. This respect of the $\star$ product is explained by studying the fuzzy algebrae of the Kaehler coset space.Comment: LaTeX, 11 pages, no figur

Fuzzy Algebrae of the General Kaehler Coset Space G/H\otimesU(1)^k

We study the fuzzy structure of the general Kaehler coset space G/S\otimes{U(1)}^k deformed by the Fedosov formalism. It is shown that the Killing potentials satisfy the fuzzy algebrae working in the Darboux coordinates.Comment: 8 pages, LaTex, no figur

The Fuzzy S^4 by Quantum Deformation

The fuzzy algebra of S^4 is discussed by quantum deformation. To this end we embed the classical S^4 in the Kaehler coset space SO(5)/U(2). The harmonic functions of S^4 are constructed in terms of the complex coordinates of SO(5)/U(2). Being endowed with the symplectic structure they can be deformed by the Fedosov formalism. We show that they generate the fuzzy algebra \hat A_\infty (S^4) under the * product defined therein, by using the Darboux coordinate system. The fuzzy spheres of higher even dimensions can be discussed similarly. We give basic arguments for the generalization as well.Comment: 20 pages, LaTex, no figur

Simpler is Much Faster: Fair and Independent Inner Product Search

The problem of inner product search (IPS) is important in many fields. Although maximum inner product search (MIPS) is often considered, its result is usually skewed and static. Users are hence hard to obtain diverse and/or new items by using the MIPS problem. Motivated by this, we formulate a new problem, namely the fair and independent IPS problem. Given a query, a threshold, and an output size k, this problem randomly samples k items from a set of items such that the inner product of the query and item is not less than the threshold. For each item that satisfies the threshold, this problem is fair, because the probability that such an item is outputted is equal to that for each other item. This fairness can yield diversity and novelty, but this problem faces a computational challenge. Some existing (M)IPS techniques can be employed in this problem, but they require O(n) or o(n) time, where n is the dataset size. To scale well to large datasets, we propose a simple yet efficient algorithm that runs in O(logn + k) expected time. We conduct experiments using real datasets, and the results demonstrate that our algorithm is up to 330 times faster than baselines.Aoyama K., Amagata D., Fujita S., et al. Simpler is Much Faster: Fair and Independent Inner Product Search. SIGIR 2023 - Proceedings of the 46th International ACM SIGIR Conference on Research and Development in Information Retrieval , 2379 (2023); https://doi.org/10.1145/3539618.3592061

The Fuzzy Kaehler Coset Space by the Fedosov Formalism

We discuss deformation quantization of the Kaehler coset space by using the Fedosov formalism. We show that the Killing potentials of the Kaehler coset space satisfy the fuzzy algebrae, when the coset space is irreducible.Comment: 12 pages, no figure, Latex; reference added and typos correcte