Multidimensional scaling is a statistical process that aims to embed high-dimensional data into a lower-dimensional, more manageable space. Common MDS algorithms tend to have some limitations when facing large data sets due to their high time and spatial complexities. This paper attempts to tackle the problem by using a stochastic approach to MDS which uses gradient descent to optimise a loss function defined on randomly designated quartets of points. This method mitigates the quadratic memory usage by computing distances on the fly, and has iterations in O(N ) time complexity, with N samples. Experiments show that the proposed method provides competitive results in reasonable time. Public codes are available at https://github.com/PierreLambert3/SQuaD-MDS.git.
Multidimensional scaling and its limitationsDimensionality reduction (DR) is the process of mapping high-dimensional (HD) observations into a lower-dimensional (LD) space such that the LD embedding is a faithful representation of the HD data. The main DR uses are in machine learning, to curb the curse of dimensionality, and in visualisation. Mapped data can reveal structures that would lay hidden from the human perception if left in HD. Typically, some information is lost by the DR and, therefore, each DR method has a take on what kind of information should be preserved and what can be lost. Used frequently in visualisation, t-SNE [1] aims at retaining the neighbourhood of each point according to a distance metric and a perplexity, which reflects the size of the neighbourhood to preserve. While t-SNE excels at retaining local structures, sufficiently remote points tend to be considered equally distant by the algorithm and, therefore, the larger-scale structures can be distorted. Such distortions can lead to erroneous conclusions by the human user, who might overestimate the dissimilarity between two clusters that are distant in the LD embedding. For this reason, using multiple DR paradigms in conjunction is a good practice in visualisation: another embedding that preserves distances instead of neighbourhoods would have prevented this erroneous conclusion.This paper considers metric multidimensional scaling (MDS): a DR technique that produces a LD embedding such that the pairwise distances in LD reflect those in HD. MDS minimises a cost function which, in its simplest form, is the sum of the squared differences between distances in HD and the Euclidean distances in LD. A common strategy to optimize this cost function is based on 417
The problem of aerodynamic shape optimization in Euler flow is addressed. B-splines are used for parame:rization of the shape. Cheap gradient calculation is obtained via sensitivity analysis and the solution of an adjoint equation; pseudo secondorder spatial accuracy is achieved by means of a semianalytical folmulation. As a validation of the approach, several inverse and constrained optimization test problems are presented with emphasis on civil engine nacelle design. The handling of nondifferentiable quantities (such as maxima) in cost functions is allowed for via the use of the Kreisselmeier-Steinhauser function.
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