Ville T. Turunen

Global quantization of pseudo-differential operators on compact Lie groups, SU(2), 3-sphere, and homogeneous spaces

Global quantization of pseudo-differential operators on general compact Lie groups G is introduced relying on the representation theory of the group rather than on expressions in local coordinates. A new class of globally defined symbols is introduced and related to the usual Hormanders classes of operators Psi(m)(G). Properties of the new class and symbolic calculus are analyzed. Properties of symbols as well as L-2-boundedness and Sobolev L-2-boundedness of operators in this global quantization are established on general compact Lie groups. Operators on the three-dimensional sphere S-3 and on group SU(2) are analyzed in detail. An application is given to pseudo-differential operators on homogeneous spaces K backslash G. In particular, using the obtained global characterization of pseudo-differential operators on Lie groups, it is shown that every pseudo-differential operator in Psi(m)(K backslash G) can be lifted to a pseudo-differential operator in Psi(m)(G), extending the known results on invariant partial differential operators.

Overview and results of Morpho challenge 2009

The goal of Morpho Challenge 2009 was to evaluate unsupervised algorithms that provide morpheme analyses for words in different languages and in various practical applications. Morpheme analysis is particularly useful in speech recognition, information retrieval and machine translation for morphologically rich languages where the amount of different word forms is very large. The evaluations consisted of: 1. a comparison to grammatical morphemes, 2. using morphemes instead of words in information retrieval tasks, and 3. combining morpheme and word based systems in statistical machine translation tasks. The evaluation languages were: Finnish, Turkish, German, English and Arabic. This paper describes the tasks, evaluation methods, and obtained results. The Morpho Challenge was part of the EU Network of Excellence PASCAL Challenge Program and organized in collaboration with CLEF.

On the Fourier Analysis of Operators on the Torus

Basic properties of Fourier integral operators on the torus \( \mathbb{T}^n = (\mathbb{R}/2\pi \mathbb{Z})^n \) are studied by using the global representations by Fourier series instead of local representations. The results can be applied in studying hyperbolic partial differential equations.

Hörmander Class of Pseudo-Differential Operators on Compact Lie Groups and Global Hypoellipticity

In this paper we give several global characterisations of the Hörmander class

Overview of Morpho challenge 2008

\Psi ^m(G)

Speech retrieval from unsegmented finnish audio using statistical morpheme-like units for segmentation, recognition, and retrieval

Ψm(G) of pseudo-differential operators on compact Lie groups in terms of the representation theory of the group. The result is applied to give criteria for the ellipticity and the global hypoellipticity of pseudo-differential operators in terms of their matrix-valued full symbols. Several examples of the first and second order globally hypoelliptic differential operators are given, in particular of operators that are locally not invertible nor hypoelliptic but globally are. Where the global hypoelliptiticy fails, one can construct explicit examples based on the analysis of the global symbols.

Morpho challenge evaluation by information retrieval experiments

On the torus, pseudo-differential operators can be presented globally by Fourier series, without local coordinate charts. Periodization of partial differential equations leads to investigating the correspondence between toroidal and Euclidean symbols of operators. Fourier series operators—the analogues of Fourier integral operators—are introduced, and formulae for their compositions with pseudo-differential operators are presented. The L 2 boundedness of pseudo-differential and Fourier series operators follows from certain conditions on phases and amplitudes.

Speech transcription and spoken document retrieval in finnish

This paper gives an overview of Morpho Challenge 2008 competition and results. The goal of the challenge was to evaluate unsupervised algorithms that provide morpheme analyses for words in different languages. For morphologically complex languages, such as Finnish, Turkish and Arabic, morpheme analysis is particularly important for lexical modeling of words in speech recognition, information retrieval and machine translation. The evaluation in Morpho Challenge competitions consisted of both a linguistic and an application oriented performance analysis. In addition to the Finnish, Turkish, German and English evaluations performed in Morpho Challenge 2007, the competition this year had an additional evaluation for Arabic. The results in linguistic evaluation in 2008 show that although the level of precision and recall varies substantially between the tasks in different languages, the best methods seem to deal quite well with all languages involved. The results in information retrieval evaluation indicate that the morpheme analysis has a significant effect in all the tested languages (Finnish, English and German). The best unsupervised and language-independent morpheme analysis methods can also rival the best language-dependent word normalization methods. The Morpho Challenge was part of the EU Network of Excellence PASCAL Challenge Program and organized in collaboration with CLEF.