Similarity metrics are a core component of many information retrieval and machine learning systems. In this work we propose a method capable of learning a similarity metric from data equipped with a binary relation. By considering only the similarity constraints, and initially ignoring the features, we are able to learn target vectors for each instance using one of several appropriately designed loss functions. A regression model can then be constructed that maps novel feature vectors to the same target vector space, resulting in a feature extractor that computes vectors for which a predefined metric is a meaningful measure of similarity. We present results on both multiclass and multi-label classification datasets that demonstrate considerably faster convergence, as well as higher accuracy on the majority of the intrinsic evaluation tasks and all extrinsic evaluation tasks.
Neighbourhood Components Analysis (NCA) @cite_11 is another method developed to be used in conjunction with @math -NN classifiers. This technique attempts to find the matrix, @math , by minimising a differentiable loss function that approximates the behaviour of @math -NN in the transformed feature space.
@cite_11
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The fused lasso, also known as total-variation denoising, is a locally-adaptive function estimator over a regular grid of design points. In this paper, we extend the fused lasso to settings in which the points do not occur on a regular grid, leading to a new approach for non-parametric regression. This approach, which we call the @math -nearest neighbors ( @math -NN) fused lasso, involves (i) computing the @math -NN graph of the design points; and (ii) performing the fused lasso over this @math -NN graph. We show that this procedure has a number of theoretical advantages over competing approaches: specifically, it inherits local adaptivity from its connection to the fused lasso, and it inherits manifold adaptivity from its connection to the @math -NN approach. We show that excellent results are obtained in a simulation study and on an application to flu data. For completeness, we also study an estimator that makes use of an @math -graph rather than a @math -NN graph, and contrast this with the @math -NN fused lasso.