We identify a fundamental issue in the popular Stochastic Neighbour Embedding (SNE and t-SNE), i.e., the "learned" similarity of any two points in high-dimensional space is not defined and cannot be computed. It underlines two previously unexplored issues in the algorithm which have undermined the quality of its final visualisation output and its ability to process large datasets. The issues are:(a) the reference probability in high-dimensional space is set based on entropy which has undefined relation with local density; and (b) the use of data independent kernel which leads to the need to determine n bandwidths for a dataset of n points. This paper establishes a principle to set the reference probability via a data-dependent kernel which has a well-defined kernel characteristic that linked directly to local density. A solution based on a recent data-dependent kernel called Isolation Kernel addresses the fundamental issue as well as its two ensuing issues. As a result, it significantly improves the quality of the final visualisation output and removes one obstacle that prevents t-SNE from processing large datasets. The solution is extremely simple, i.e., simply replacing the existing data independent kernel with Isolation Kernel, leaving the rest of the t-SNE procedure unchanged.
There are some works focusing on analysing the heuristics methods for solving non-convex optimisation problems for the embedding . Recently, @cite_5 theoretically analyse this optimisation and provide a framework to make clusterable data visually identifiable in the 2-dimensional embedding space. These works are not related to similarity measurements; therefore not directly relevant to work reported here.
@cite_5
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A first line of attack in exploratory data analysis is data visualization, i.e., generating a 2-dimensional representation of data that makes clusters of similar points visually identifiable. Standard Johnson-Lindenstrauss dimensionality reduction does not produce data visualizations. The t-SNE heuristic of van der Maaten and Hinton, which is based on non-convex optimization, has become the de facto standard for visualization in a wide range of applications. This work gives a formal framework for the problem of data visualization - finding a 2-dimensional embedding of clusterable data that correctly separates individual clusters to make them visually identifiable. We then give a rigorous analysis of the performance of t-SNE under a natural, deterministic condition on the "ground-truth" clusters (similar to conditions assumed in earlier analyses of clustering) in the underlying data. These are the first provable guarantees on t-SNE for constructing good data visualizations. We show that our deterministic condition is satisfied by considerably general probabilistic generative models for clusterable data such as mixtures of well-separated log-concave distributions. Finally, we give theoretical evidence that t-SNE provably succeeds in partially recovering cluster structure even when the above deterministic condition is not met. Low-rank representation (LRR) intends to find the representation with lowest rank of a given data set, which can be formulated as a rank-minimisation problem. Since the rank operator is non-convex and discontinuous, most of the recent works use the nuclear norm as a convex relaxation. It is theoretically shown that, under some conditions, the Frobenius-norm-based optimisation problem has a unique solution that is also a solution of the original LRR optimisation problem. In other words, it is feasible to apply the Frobenius norm as a surrogate of the non-convex matrix rank function. This replacement will largely reduce the time costs for obtaining the lowest-rank solution. Experimental results show that the method (i.e. fast LRR (fLRR)) performs well in terms of accuracy and computation speed in image clustering and motion segmentation compared with nuclear-norm-based LRR algorithm. In this work we introduce a new optimisation method called SAGA in the spirit of SAG, SDCA, MISO and SVRG, a set of recently proposed incremental gradient algorithms with fast linear convergence rates. SAGA improves on the theory behind SAG and SVRG, with better theoretical convergence rates, and has support for composite objectives where a proximal operator is used on the regulariser. Unlike SDCA, SAGA supports non-strongly convex problems directly, and is adaptive to any inherent strong convexity of the problem. We give experimental results showing the effectiveness of our method. Randomised search heuristics are used in practice to solve difficult problems where no good problem-specific algorithm is known. They deliver a solution of acceptable quality in reasonable time in many cases. When theoretically analysing the performance of randomised search heuristics one usually considers the average time needed to find an optimal solution or one of a pre-specified approximation quality. This is very different from practice where usually the algorithm is stopped after some time. For a theoretical analysis this corresponds to investigating the quality of the solution obtained after a pre-specified number of function evaluations called budget. Such a perspective is taken here and two simple randomised search heuristics, random local search and the (1+1) evolutionary algorithm, are analysed on simple and well-known example functions. If the budget is significantly smaller than the expected time needed for optimisation the behaviour of the algorithms can be very different depending on the problem at hand. Precise analytical results are proven. They demonstrate novel and interesting challenges in the analysis of randomised search heuristics. The potential of this different perspective to provide a more practically useful theory is shown. Kernel-based learning algorithms work by embedding the data into a Euclidean space, and then searching for linear relations among the embedded data points. The embedding is performed implicitly, by specifying the inner products between each pair of points in the embedding space. This information is contained in the so-called kernel matrix, a symmetric and positive semidefinite matrix that encodes the relative positions of all points. Specifying this matrix amounts to specifying the geometry of the embedding space and inducing a notion of similarity in the input space---classical model selection problems in machine learning. In this paper we show how the kernel matrix can be learned from data via semidefinite programming (SDP) techniques. When applied to a kernel matrix associated with both training and test data this gives a powerful transductive algorithm---using the labeled part of the data one can learn an embedding also for the unlabeled part. The similarity between test points is inferred from training points and their labels. Importantly, these learning problems are convex, so we obtain a method for learning both the model class and the function without local minima. Furthermore, this approach leads directly to a convex method for learning the 2-norm soft margin parameter in support vector machines, solving an important open problem. We consider two questions arising in the analysis of heuristic algorithms. (i) Is there a general procedure involved when analysing a particular problem heuristic? (ii) How can heuristic procedures be incorporated into optimising algorithms such as branch and bound? Approaches combining genetic algorithms and neural networks have received a great deal of attention in recent years. As a result, much work has been reported in two major areas of neural network design: training and topology optimisation. This paper focuses on the key issues associated with the problem of pruning a multilayer perceptron using genetic algorithms and simulated annealing. The study presented considers a number of aspects associated with network training that may alter the behaviour of a stochastic topology optimiser. Enhancements are discussed that can improve topology searches. Simulation results for the two mentioned stochastic optimisation methods applied to non-linear system identification are presented and compared with a simple random search. We present a novel method for constructing a visual vocabulary that takes into account the class labels of images, thus resulting in better recognition performance and more efficient learning. Our method consists of two stages: Cluster Precision Maximisation (CPM) and Adaptive Refinement. In the first stage, a Reciprocal Nearest Neighbours (RNN) clustering algorithm is guided towards class representative visual words by maximising a new cluster precision criterion. As we are able to optimise the vocabulary without the need for expensive cross-validation, the overall training time is significantly reduced without a negative impact on the results. Next, an adaptive threshold refinement scheme is proposed with the aim of increasing vocabulary compactness while at the same time improving the recognition rate and further increasing the representativeness of the visual words for category-level object recognition. This is a correlation clustering based approach, which works as a meta-clustering and optimises the cut-off threshold for each cluster separately. In the experiments we analyse the recognition rate of different vocabularies for a subset of the Caltech 101 dataset, showing how RNN in combination with CPM selects the optimal codebooks, and how the clustering refinement step succeeds in further increasing the recognition rate. In this paper, we study the problem of dynamic user profiling in Twitter. We address the problem by proposing a dynamic user and word embedding model (DUWE), a scalable black-box variational inference algorithm, and a streaming keyword diversification model (SKDM). DUWE dynamically tracks the semantic representations of users and words over time and models their embeddings in the same space so that their similarities can be effectively measured. Our inference algorithm works with a convex objective function that ensures the robustness of the learnt embeddings. SKDM aims at retrieving top-K relevant and diversified keywords to profile users' dynamic interests. Experiments on a Twitter dataset demonstrate that our proposed embedding algorithms outperform state-of-the-art non-dynamic and dynamic embedding and topic models. Neural Network approaches to time series prediction are briefly discussed, and the need to find the appropriate sample rate and an appropriately sized input window identified. Relevant theoretical results from dynamic systems theory are briefly introduced, and heuristics for finding the appropriate sampling rate and embedding dimension, and thence window size, are discussed. The method is applied to several time series and the resulting generalisation performance of the trained feed-forward neural network predictors is analysed. It is shown that the heuristics can provide useful information in defining the appropriate network architecture. Secure multi-party computation has been considered by the cryptographic community for a number of years. Until recently it has been a purely theoretical area, with few implementations with which to test various ideas. This has led to a number of optimisations being proposed which are quite restricted in their application. In this paper we describe an implementation of the two-party case, using Yao's garbled circuits, and present various algorithmic protocol improvements. These optimisations are analysed both theoretically and empirically, using experiments of various adversarial situations. Our experimental data is provided for reasonably large circuits, including one which performs an AES encryption, a problem which we discuss in the context of various possible applications. Arguably the greatest single work in the history of psychology. James's analyses of habit, the nature of emotion, the phenomenology of attention, the stream of thought, the perception of space, and the multiplicity of the consciousness of self are still widely cited and incorporated into contemporary theoretical accounts of these phenomena. This article challenges recent research (Evans, 2008) reporting that the concentration of cited scientific literature increases with the online availability of articles and journals. Using Thomson Reuters' Web of Science, the present article analyses changes in the concentration of citations received (2- and 5-year citation windows) by papers published between 1900 and 2005. Three measures of concentration are used: the percentage of papers that received at least one citation (cited papers); the percentage of papers needed to account for 20 , 50 , and 80 of the citations; and the Herfindahl-Hirschman index (HHI). These measures are used for four broad disciplines: natural sciences and engineering, medical fields, social sciences, and the humanities. All these measures converge and show that, contrary to what was reported by Evans, the dispersion of citations is actually increasing. © 2009 Wiley Periodicals, Inc. Automatic classification of activities in a sports arena is important in order to analyse and optimise the use of the arenas. In this work we classify five sports types based only on occupancy heatmaps produced from position data. Due to privacy issues we use thermal imaging for detecting people and then calculate their positions on the court using homography. Heatmaps are produced by summarising Gaussian distributions respresenting people over 10-minute periods. Before classification the heatmaps are projected to a low-dimensional discriminative space using the principle of Fisherfaces. Our result using two weeks of video are very promising with a correct classification of 90.76 . Deep learning has achieved great success in face recognition, however deep-learned features still have limited invariance to strong intra-personal variations such as large pose changes. It is observed that some facial attributes (e.g. eyebrow thickness, gender) are robust to such variations. We present the first work to systematically explore how the fusion of face recognition features (FRF) and facial attribute features (FAF) can enhance face recognition performance in various challenging scenarios. Despite the promise of FAF, we find that in practice existing fusion methods fail to leverage FAF to boost face recognition performance in some challenging scenarios. Thus, we develop a powerful tensor-based framework which formulates feature fusion as a tensor optimisation problem. It is nontrivial to directly optimise this tensor due to the large number of parameters to optimise. To solve this problem, we establish a theoretical equivalence between low-rank tensor optimisation and a two-stream gated neural network. This equivalence allows tractable learning using standard neural network optimisation tools, leading to accurate and stable optimisation. Experimental results show the fused feature works better than individual features, thus proving for the first time that facial attributes aid face recognition. We achieve state-of-the-art performance on three popular databases: MultiPIE (cross pose, lighting and expression), CASIA NIR-VIS2.0 (cross-modality environment) and LFW (uncontrolled environment). Embedding complex objects as vectors in low dimensional spaces is a longstanding problem in machine learning. We propose in this work an extension of that approach, which consists in embedding objects as elliptical probability distributions, namely distributions whose densities have elliptical level sets. We endow these measures with the 2-Wasserstein metric, with two important benefits: For such measures, the squared 2-Wasserstein metric has a closed form, equal to the sum of the squared Euclidean distance between means and the squared Bures metric between covariance matrices. The latter is a Riemannian metric between positive semi-definite matrices, which turns out to be Euclidean on a suitable factor representation of such matrices, which is valid on the entire geodesic between these matrices. The 2-Wasserstein distance boils down to the usual Euclidean metric when comparing Diracs, and therefore provides the natural framework to extend point embeddings. We show that for these reasons Wasserstein elliptical embeddings are more intuitive and yield tools that are better behaved numerically than the alternative choice of Gaussian embeddings with the Kullback-Leibler divergence. In particular, and unlike previous work based on the KL geometry, we learn elliptical distributions that are not necessarily diagonal. We demonstrate the interest of elliptical embeddings by using them for visualization, to compute embeddings of words, and to reflect entanglement or hypernymy. Abstract We describe a method for recovering the underlying parametrization of scattered data (mi) lying on a manifold M embedded in high-dimensional Euclidean space. The method, Hessian-based locally linear embedding, derives from a conceptual framework of local isometry in which the manifold M, viewed as a Riemannian submanifold of the ambient Euclidean space ℝn, is locally isometric to an open, connected subset Θ of Euclidean space ℝd. Because Θ does not have to be convex, this framework is able to handle a significantly wider class of situations than the original ISOMAP algorithm. The theoretical framework revolves around a quadratic form ℋ(f) = ∫M ∥Hf(m)∥dm defined on functions f : M ↦ ℝ. Here Hf denotes the Hessian of f, and ℋ(f) averages the Frobenius norm of the Hessian over M. To define the Hessian, we use orthogonal coordinates on the tangent planes of M. The key observation is that, if M truly is locally isometric to an open, connected subset of ℝd, then ℋ(f) has a (d + 1)-dimensional null space consisting of the constant functions and a d-dimensional space of functions spanned by the original isometric coordinates. Hence, the isometric coordinates can be recovered up to a linear isometry. Our method may be viewed as a modification of locally linear embedding and our theoretical framework as a modification of the Laplacian eigenmaps framework, where we substitute a quadratic form based on the Hessian in place of one based on the Laplacian. The different factors involved in the growth process of complex networks imprint valuable information in their observable topologies. How to exploit this information to accurately predict structural network changes is the subject of active research. A recent model of network growth sustains that the emergence of properties common to most complex systems is the result of certain trade-offs between node birth-time and similarity. This model has a geometric interpretation in hyperbolic space, where distances between nodes abstract this optimisation process. Current methods for network hyperbolic embedding search for node coordinates that maximise the likelihood that the network was produced by the afore-mentioned model. Here, a different strategy is followed in the form of the Laplacian-based Network Embedding, a simple yet accurate, efficient and data driven manifold learning approach, which allows for the quick geometric analysis of big networks. Comparisons against existing embedding and prediction techniques highlight its applicability to network evolution and link prediction. Background The protein folding problem is a fundamental problems in computational molecular biology and biochemical physics. Various optimisation methods have been applied to formulations of the ab-initio folding problem that are based on reduced models of protein structure, including Monte Carlo methods, Evolutionary Algorithms, Tabu Search and hybrid approaches. In our work, we have introduced an ant colony optimisation (ACO) algorithm to address the non-deterministic polynomial-time hard (NP-hard) combinatorial problem of predicting a protein's conformation from its amino acid sequence under a widely studied, conceptually simple model – the 2-dimensional (2D) and 3-dimensional (3D) hydrophobic-polar (HP) model. Clustering is central to many data-driven application domains and has been studied extensively in terms of distance functions and grouping algorithms. Relatively little work has focused on learning representations for clustering. In this paper, we propose Deep Embedded Clustering (DEC), a method that simultaneously learns feature representations and cluster assignments using deep neural networks. DEC learns a mapping from the data space to a lower-dimensional feature space in which it iteratively optimizes a clustering objective. Our experimental evaluations on image and text corpora show significant improvement over state-of-the-art methods.