Dimensionality reduction is a fundamental task in modern data science. Several projection methods specifically tailored to take into account the non-linearity of the data via local embeddings have been proposed. Such methods are often based on local neighbourhood structures and require tuning the number of neighbours that define this local structure, and the dimensionality of the lower-dimensional space onto which the data are projected. Such choices critically influence the quality of the resulting embedding. In this paper, we exploit a recently proposed intrinsic dimension estimator which also returns the optimal locally adaptive neighbourhood sizes according to some desirable criteria. In principle, this adaptive framework can be employed to perform an optimal hyper-parameter tuning of any dimensionality reduction algorithm that relies on local neighbourhood structures. Numerical experiments on both real-world and simulated datasets show that the proposed method can be used to significantly improve well-known projection methods when employed for various learning tasks, with improvements measurable through both quantitative metrics and the quality of low-dimensional visualizations.

We thank the reviewers for their positive feedback on our paper. Rev 2 and 5 asked the difference between LL-LVM and GP-LVM: As noted in lines 65-68 in introduction, both are Gaussian models, but GP-LVM is parameterised by a stationary covariance function rather than by a local neighbour-defined precision matrix which seeks to explicitly preserve local neighbourhood properties. Rev 1 We will add the complexity analysis. An expensive step in the E-step is the inversion of the posterior precision matrices of both x and C (in eq. We will study possible ways to decrease the complexity in future work.

Local search is a common method for solving combinatorial optimisation problems. We focus on general-purpose local search solvers that accept as input a constraint model -- a declarative description of a problem consisting of a set of decision variables under a set of constraints. Existing approaches typically take as input models written in solver-independent constraint modelling languages like MiniZinc. The Athanor solver we describe herein differs in that it begins from a specification of a problem in the abstract constraint specification language Essence, which allows problems to be described without commitment to low-level modelling decisions through its support for a rich set of abstract types. The advantage of proceeding from Essence is that the structure apparent in a concise, abstract specification of a problem can be exploited to generate high quality neighbourhoods automatically, avoiding the difficult task of identifying that structure in an equivalent constraint model. Based on the twin benefits of neighbourhoods derived from high level types and the scalability derived by searching directly over those types, our empirical results demonstrate strong performance in practice relative to existing solution methods.

Distance metric learning (DML) approaches learn a transformation to a representation space where distance is in correspondence with a predefined notion of similarity. While such models offer a number of compelling benefits, it has been difficult for these to compete with modern classification algorithms in performance and even in feature extraction. In this work, we propose a novel approach explicitly designed to address a number of subtle yet important issues which have stymied earlier DML algorithms. It maintains an explicit model of the distributions of the different classes in representation space. It then employs this knowledge to adaptively assess similarity, and achieve local discrimination by penalizing class distribution overlap. We demonstrate the effectiveness of this idea on several tasks. Our approach achieves state-of-the-art classification results on a number of fine-grained visual recognition datasets, surpassing the standard softmax classifier and outperforming triplet loss by a relative margin of 30-40%. In terms of computational performance, it alleviates training inefficiencies in the traditional triplet loss, reaching the same error in 5-30 times fewer iterations. Beyond classification, we further validate the saliency of the learnt representations via their attribute concentration and hierarchy recovery properties, achieving 10-25% relative gains on the softmax classifier and 25-50% on triplet loss in these tasks.

We propose a probabilistic model to infer supervised latent variables in the Hamming space from observed data. Our model allows simultaneous inference of the number of binary latent variables, and their values. The latent variables preserve neighbourhood structure of the data in a sense that objects in the same semantic concept have similar latent values, and objects in different concepts have dissimilar latent values. We formulate the supervised infinite latent variable problem based on an intuitive principle of pulling objects together if they are of the same type, and pushing them apart if they are not. We then combine this principle with a flexible Indian Buffet Process prior on the latent variables. We show that the inferred supervised latent variables can be directly used to perform a nearest neighbour search for the purpose of retrieval. We introduce a new application of dynamically extending hash codes, and show how to effectively couple the structure of the hash codes with continuously growing structure of the neighbourhood preserving infinite latent feature space.

This paper is concerned with local search methods to solve job shop scheduling problems with uncertain durations modelled as fuzzy numbers. Based on a neighbourhood structure from the literature, a reduced set of moves and the consequent structure are defined. Theoretical results show that the proposed neighbourhood contains all the improving solutions from the original neighbourhood and provide a sufficient condition for optimality. Additionally, a makespan lower bound is proposed which can be used to discard neighbours. Experimental results illustrate the good performance of both proposals, which considerably reduce the computational load of the local search, as well as a synergy effect when they are simultaneously used.