Analysis of Large-Scale mRNA Expression Data Sets by Genetic Algorithms
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The intrinsic dimensionality refers to the "true" dimensionality of the data, as opposed to the dimensionality of the data representation. For example, when attributes are highly correlated, the intrinsic dimensionality can be much lower than the number of variables. Local intrinsic dimensionality refers to the observation that this property can vary for different parts of the data set; and intrinsic dimensionality can serve as a proxy for the local data complexity of the data set. Most popular methods for estimating the local intrinsic dimensionality are based on distances, and the rate at which the distances to the nearest neighbors increase, a concept known as "expansion dimension". In this paper we introduce an orthogonal concept, which does not use any distances: we use the distribution of angles among the neighbors with respect to the query point. We derive the theoretical distribution of angles and use this to construct an estimator for intrinsic dimensionality. We derive a regularized version that obeys an upper bound of the true intrinsic dimensionality for idealized data. We generalize the estimator to arbitrary moments and discuss the relation of the regularized second moment to approaches based on principal component analysis. Experimentally, we verify that this measure behaves similarly but complementarily to existing measures of intrinsic dimensionality. By introducing a new idea of intrinsic dimensionality to the research community we hope to contribute to a better understanding of intrinsic dimensionality and to spur new research in this direction.
Intrinsic dimension
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Multidimensional scaling
Matrix (chemical analysis)
Transfer matrix
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Whether the theoretical innovation is realized or not is related to the knowledge dimensionality, the value dimensionality and the thinking dimensionality involved in theory, and it also depends on the unity of the truth, the good and the new conception of the three dimensionalities of theory . The knowledge dimensionality, determining the possibility of the theoretical innovation, is considered as the prerequisite and the base. The value dimensionality, determining the direction of the theoretical innovation , is used as the motive force. Meanwhile the thinking dimensionality, being a decisive factor for the styles and means of the theoretical innovation, is the guarantee .
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Low-dimensional materials have excellent properties which are closely related to their dimensionality. However, the growth mechanism underlying tunable dimensionality from 2D triangles to ID ribbons of such materials is still unrevealed. Here, we establish a general kinetic Monte Carlo model for transition metal dichalcogenides (TMDs) growth to address such an issue. Our model is able to reproduce several key ñndings in experiments, and reveals that the dimensionality is determined by the lattice mismatch and the interaction strength between TMDs and the substrate. We predict that the dimensionality can be well tuned by the interaction strength and the geometry of the substrate. Our work deepens the understanding of tunable dimensionality of low-dimensional materials and may inspire new concepts for the design of such materials with expected dimensionality.
Lattice (music)
Kinetic Monte Carlo
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Liquid phase
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It is well known that metaheuristics for numerical optimization tend to decrease in performance as dimensionality increases. These effects are commonly referred to as "The Curse of Dimensionality". An obvious change to search spaces with increasing dimensionality is that their volume grows exponentially, and this has led to large amounts of research on improved exploration. A recent insight is that the shape of attraction basins can also change drastically with increasing dimensionality, and this has led to selection-based approaches to combat the Curse of Dimensionality. Average-Fitness Based Selection is introduced as a means to reduce the selection errors caused by Fitness-Based Selection. Experimental results show that the rate of selection errors grows much more slowly for Average-Fitness Based Selection with Increasing dimensionality.
Winner's curse
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In this paper, we establish a neural network to approximate functionals, which are maps from infinite dimensional spaces to finite dimensional spaces.The approximation error of the neural network is O(1/ √ m) where m is the size of networks, which overcomes the curse of dimensionality.The key idea of the approximation is to define a Barron space of functionals.
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We report on a novel use of parallel coordinates as a pedagogical tool for illustrating the non-intuitive properties of high dimensional spaces with special emphasis on the phenomenon of Curse of Dimensionality. Also, we have collated what we believe to be a representative sample of diverse approaches that exist in literature to conceptualize the Curse of Dimensionality. We envisage that the paper will have pedagogical value in structuring the way Curse of Dimensionality is presented in classrooms and associated lab sessions.
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