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
2142040002 2950254505 2199495299 2098620241 2108282816 1968265719 2036959865 2397509712 1541459201 2963687412 2147880780 85455704 2952237618 2111575001 2125742596 2953167455 2149908819 2044520258 2103844245 1916116059
Recent advances in computer vision and machine learning suggest that a wide range of problems can be addressed more appropriately by considering non-Euclidean geometry. In this paper we explore sparse dictionary learning over the space of linear subspaces, which form Riemannian structures known as Grassmann manifolds. To this end, we propose to embed Grassmann manifolds into the space of symmetric matrices by an isometric mapping, which enables us to devise a closed-form solution for updating a Grassmann dictionary, atom by atom. Furthermore, to handle non-linearity in data, we propose a kernelised version of the dictionary learning algorithm. Experiments on several classification tasks (face recognition, action recognition, dynamic texture classification) show that the proposed approach achieves considerable improvements in discrimination accuracy, in comparison to state-of-the-art methods such as kernelised Affine Hull Method and graph-embedding Grassmann discriminant analysis. Recent advances in computer vision and machine learning suggest that a wide range of problems can be addressed more appropriately by considering non-Euclidean geometry. In this paper we explore sparse dictionary learning over the space of linear subspaces, which form Riemannian structures known as Grassmann manifolds. To this end, we propose to embed Grassmann manifolds into the space of symmetric matrices by an isometric mapping, which enables us to devise a closed-form solution for updating a Grassmann dictionary, atom by atom. Furthermore, to handle non-linearity in data, we propose a kernelised version of the dictionary learning algorithm. Experiments on several classification tasks (face recognition, action recognition, dynamic texture classification) show that the proposed approach achieves considerable improvements in discrimination accuracy, in comparison to state-of-the-art methods such as kernelised Affine Hull Method and graph-embedding Grassmann discriminant analysis. In k-means clustering we are given a set of n data points in d-dimensional space Rd and an integer k, and the problem is to determine a set of k points in &#211C;d, called centers, to minimize the mean squared distance from each data point to its nearest center. No exact polynomial-time algorithms are known for this problem. Although asymptotically efficient approximation algorithms exist, these algorithms are not practical due to the extremely high constant factors involved. There are many heuristics that are used in practice, but we know of no bounds on their performance.We consider the question of whether there exists a simple and practical approximation algorithm for k-means clustering. We present a local improvement heuristic based on swapping centers in and out. We prove that this yields a (9+e)-approximation algorithm. We show that the approximation factor is almost tight, by giving an example for which the algorithm achieves an approximation factor of (9-e). To establish the practical value of the heuristic, we present an empirical study that shows that, when combined with Lloyd's algorithm, this heuristic performs quite well in practice. Visualization methods that arrange data objects in 2D or 3D layouts have followed two main schools, methods oriented for graph layout and methods oriented for vectorial embedding. We show the two previously separate approaches are tied by an optimization equivalence, making it possible to relate methods from the two approaches and to build new methods that take the best of both worlds. In detail, we prove a theorem of optimization equivalences between β- and γ-, as well as α- and Renyi-divergences through a connection scalar. Through the equivalences we represent several nonlinear dimensionality reduction and graph drawing methods in a generalized stochastic neighbor embedding setting, where information divergences are minimized between similarities in input and output spaces, and the optimal connection scalar provides a natural choice for the tradeoff between attractive and repulsive forces. We give two examples of developing new visualization methods through the equivalences: 1) We develop weighted symmetric stochastic neighbor embedding (ws-SNE) from Elastic Embedding and analyze its benefits, good performance for both vectorial and network data; in experiments ws-SNE has good performance across data sets of different types, whereas comparison methods fail for some of the data sets; 2) we develop a γ-divergence version of a PolyLog layout method; the new method is scale invariant in the output space and makes it possible to efficiently use large-scale smoothed neighborhoods. We present a novel linear clustering framework (DIFFRAC) which relies on a linear discriminative cost function and a convex relaxation of a combinatorial optimization problem. The large convex optimization problem is solved through a sequence of lower dimensional singular value decompositions. This framework has several attractive properties: (1) although apparently similar to K-means, it exhibits superior clustering performance than K-means, in particular in terms of robustness to noise. (2) It can be readily extended to non linear clustering if the discriminative cost function is based on positive definite kernels, and can then be seen as an alternative to spectral clustering. (3) Prior information on the partition is easily incorporated, leading to state-of-the-art performance for semi-supervised learning, for clustering or classification. We present empirical evaluations of our algorithms on synthetic and real medium-scale datasets. Low-dimensional embedding based on non-metric data (e.g., non-metric multidimensional scaling) is a problem that arises in many applications, especially those involving human subjects. This paper investigates the problem of learning an embedding of n objects into d-dimensional Euclidean space that is consistent with pairwise comparisons of the type “object a is closer to object b than c.” While there are O(n3) such comparisons, experimental studies suggest that relatively few are necessary to uniquely determine the embedding up to the constraints imposed by all possible pairwise comparisons (i.e., the problem is typically over-constrained). This paper is concerned with quantifying the minimum number of pairwise comparisons necessary to uniquely determine an embedding up to all possible comparisons. The comparison constraints stipulate that, with respect to each object, the other objects are ranked relative to their proximity. We prove that at least Q(dn log n) pairwise comparisons are needed to determine the embedding of all n objects. The lower bounds cannot be achieved by using randomly chosen pairwise comparisons. We propose an algorithm that exploits the low-dimensional geometry in order to accurately embed objects based on relatively small number of sequentially selected pairwise comparisons and demonstrate its performance with experiments. We study the metric properties of finite subsets of L1. The analysis of such metrics is central to a number of important algorithmic problems involving the cut structure of weighted graphs, including the Sparsest Cut Problem, one of the most compelling open problems in the field of approximation algorithms. Additionally, many open questions in geometric non-linear functional analysis involve the properties of finite subsets of L1.We present some new observations concerning the relation of L1 to dimension, topology, and Euclidean distortion. We show that every n-point subset of L1 embeds into L2 with average distortion O(√log n), yielding the first evidence that the conjectured worst-case bound of O(√log n) is valid. We also address the issue of dimension reduction in Lp for p ∈ (1, 2). We resolve a question left open by M. Charikar and A. Sahai [Dimension reduction in the l1 norm, in: Proceedings of the 43rd Annual IEEE Conference on Foundations of Computer Science, ACM, 2002, pp. 251-260] concerning the impossibility of dimension reduction with a linear map in the above cases, and we show that a natural variant of the recent example of Brinkman and Charikar [On the impossibility of dimension reduction in l1, in: Proceedings of the 44th Annual IEEE Conference on Foundations of Computer Science, ACM, 2003, pp. 514-523], cannot be used to prove a lower bound for the non-linear case. This is acomplished by exhibiting constant-distortion embeddings of snowflaked planar metrics into Euclidean space. A novel multi-criteria optimization framework for matching of partially visible shapes in multiple images using joint geometric graph embedding is proposed. The proposed framework achieves matching of partial shapes in images that exhibit extreme variations in scale, orientation, viewpoint and illumination and also instances of occlusion; conditions which render impractical the use of global contour-based descriptors or local pixel-level features for shape matching. The proposed technique is based on optimization of the embedding distances of geometric features obtained from the eigenspectrum of the joint image graph, coupled with regularization over values of the mean pixel intensity or histogram of oriented gradients. It is shown to obtain successfully the correspondences denoting partial shape similarities as well as correspondences between feature points in the images. A new benchmark dataset is proposed which contains disparate image pairs with extremely challenging variations in viewing conditions when compared to an existing dataset [18]. The proposed technique is shown to significantly outperform several state-of-the-art partial shape matching techniques on both datasets. For similarity search in high-dimensional vector spaces (or ‘HDVSs’), researchers have proposed a number of new methods (or adaptations of existing methods) based, in the main, on data-space partitioning. However, the performance of these methods generally degrades as dimensionality increases. Although this phenomenon-known as the ‘dimensional curse’-is well known, little or no quantitative a.nalysis of the phenomenon is available. In this paper, we provide a detailed analysis of partitioning and clustering techniques for similarity search in HDVSs. We show formally that these methods exhibit linear complexity at high dimensionality, and that existing methods are outperformed on average by a simple sequential scan if the number of dimensions exceeds around 10. Consequently, we come up with an alternative organization based on approximations to make the unavoidable sequential scan as fast as possible. We describe a simple vector approximation scheme, called VA-file, and report on an experimental evaluation of this and of two tree-based index methods (an R*-tree and an X-tree). Author(s): Anandkumar, A; Ge, R | Abstract: Local search heuristics for non-convex optimizations are popular in applied machine learning. However, in general it is hard to guarantee that such algorithms even converge to a local minimum, due to the existence of complicated saddle point structures in high dimensions. Many functions have degenerate saddle points such that the first and second order derivatives cannot distinguish them with local optima. In this paper we use higher order derivatives to escape these saddle points: we design the first efficient algorithm guaranteed to converge to a third order local optimum (while existing techniques are at most second order). We also show that it is NP-hard to extend this further to finding fourth order local optima. We investigate techniques for analysis and retrieval of object trajectories in two or three dimensional space. Such data usually contain a large amount of noise, that has made previously used metrics fail. Therefore, we formalize non-metric similarity functions based on the longest common subsequence (LCSS), which are very robust to noise and furthermore provide an intuitive notion of similarity between trajectories by giving more weight to similar portions of the sequences. Stretching of sequences in time is allowed, as well as global translation of the sequences in space. Efficient approximate algorithms that compute these similarity measures are also provided. We compare these new methods to the widely used Euclidean and time warping distance functions (for real and synthetic data) and show the superiority of our approach, especially in the strong presence of noise. We prove a weaker version of the triangle inequality and employ it in an indexing structure to answer nearest neighbor queries. Finally, we present experimental results that validate the accuracy and efficiency of our approach. We consider the non-metric multidimensional scaling problem: given a set of dissimilarities ∆, find an embedding whose inter-point Euclidean distances have the same ordering as ∆. In this paper, we look at a generalization of this problem in which only a set of order relations of the form dij < dkl are provided. Unlike the original problem, these order relations can be contradictory and need not be specified for all pairs of dissimilarities. We argue that this setting is more natural in some experimental settings and propose an algorithm based on convex optimization techniques to solve this problem. We apply this algorithm to human subject data from a psychophysics experiment concerning how reflectance properties are perceived. We also look at the standard NMDS problem, where a dissimilarity matrix ∆ is provided as input, and show that we can always find an orderrespecting embedding of ∆. Graph matching---aligning a pair of graphs to minimize their edge disagreements---has received wide-spread attention from both theoretical and applied communities over the past several decades, including combinatorics, computer vision, and connectomics. Its attention can be partially attributed to its computational difficulty. Although many heuristics have previously been proposed in the literature to approximately solve graph matching, very few have any theoretical support for their performance. A common technique is to relax the discrete problem to a continuous problem, therefore enabling practitioners to bring gradient-descent-type algorithms to bear. We prove that an indefinite relaxation (when solved exactly) almost always discovers the optimal permutation, while a common convex relaxation almost always fails to discover the optimal permutation. These theoretical results suggest that initializing the indefinite algorithm with the convex optimum might yield improved practical performance. Indeed, experimental results illuminate and corroborate these theoretical findings, demonstrating that excellent results are achieved in both benchmark and real data problems by amalgamating the two approaches. When for a difficult real-world optimisation problem no good problem-specific algorithm is available often randomised search heuristics are used. They are hoped to deliver good solutions in acceptable time. The theoretical analysis usually concentrates on the average time needed to find an optimal or approximately optimal solution. This matches neither the application in practice nor the empirical analysis since usually optimal solutions are not known and even if found cannot be recognised. More often the algorithms are stopped after some time. This motivates a theoretical analysis to concentrate on the quality of the best solution obtained after a pre-specified number of function evaluations called budget. Using this perspective two simple randomised search heuristics, random local search and the (1+1) evolutionary algorithm, are analysed on some well-known example problems. Upper and lower bounds on the expected quality of a solution for a fixed budget of function evaluations are proven. The analysis shows novel and challenging problems in the study of randomised search heuristics. It demonstrates the potential of this shift in perspective from expected run time to expected solution quality. We introduce two appearance-based methods for clustering a set of images of 3D (three-dimensional) objects, acquired under varying illumination conditions, into disjoint subsets corresponding to individual objects. The first algorithm is based on the concept of illumination cones. According to the theory, the clustering problem is equivalent to finding convex polyhedral cones in the high-dimensional image space. To efficiently determine the conic structures hidden in the image data, we introduce the concept of conic affinity, which measures the likelihood of a pair of images belonging to the same underlying polyhedral cone. For the second method, we introduce another affinity measure based on image gradient comparisons. The algorithm operates directly on the image gradients by comparing the magnitudes and orientations of the image gradient at each pixel. Both methods have clear geometric motivations, and they operate directly on the images without the need for feature extraction or computation of pixel statistics. We demonstrate experimentally that both algorithms are surprisingly effective in clustering images acquired under varying illumination conditions with two large, well-known image data sets. A new result in convex analysis on the calculation of proximity operators in certain scaled norms is derived. We describe efficient implementations of the proximity calculation for a useful class of functions; the implementations exploit the piece-wise linear nature of the dual problem. The second part of the paper applies the previous result to acceleration of convex minimization problems, and leads to an elegant quasi-Newton method. The optimization method compares favorably against state-of-the-art alternatives. The algorithm has extensive applications including signal processing, sparse recovery and machine learning and classification. Clustering performance can often be greatly improved by leveraging side information. In this paper, we consider constrained clustering with pairwise constraints, which specify some pairs of objects from the same cluster or not. The main idea is to design a kernel to respect both the proximity structure of the data and the given pairwise constraints. We propose a spectral kernel learning framework and formulate it as a convex quadratic program, which can be optimally solved efficiently. Our framework enjoys several desirable features: 1) it is applicable to multi-class problems; 2) it can handle both must-link and cannot-link constraints; 3) it can propagate pairwise constraints effectively; 4) it is scalable to large-scale problems; and 5) it can handle weighted pairwise constraints. Extensive experiments have demonstrated the superiority of the proposed approach. We introduce a novel framework for nonrigid feature matching among multiple sets in a way that takes into consideration both the feature descriptor and the features spatial arrangement. We learn an embedded representation that combines both the descriptor similarity and the spatial arrangement in a unified Euclidean embedding space. This unified embedding is reached by minimizing an objective function that has two sources of weights; the feature spatial arrangement and the feature descriptor similarity scores across the different sets. The solution can be obtained directly by solving one Eigen-value problem that is linear in the number of features. Therefore, the framework is very efficient and can scale up to handle a large number of features. Experimental evaluation is done using different sets showing outstanding results compared to the state of the art; up to 100 accuracy is achieved in the case of the well known ‘Hotel’ sequence. Until now, neighbor-embedding-based (NE) algorithms for super-resolution (SR) have carried out two independent processes to synthesize high-resolution (HR) image patches. In the first process, neighbor search is performed using the Euclidean distance metric, and in the second process, the optimal weights are determined by solving a constrained least squares problem. However, the separate processes are not optimal. In this paper, we propose a sparse neighbor selection scheme for SR reconstruction. We first predetermine a larger number of neighbors as potential candidates and develop an extended Robust-SL0 algorithm to simultaneously find the neighbors and to solve the reconstruction weights. Recognizing that the k-nearest neighbor (k-NN) for reconstruction should have similar local geometric structures based on clustering, we employ a local statistical feature, namely histograms of oriented gradients (HoG) of low-resolution (LR) image patches, to perform such clustering. By conveying local structural information of HoG in the synthesis stage, the k-NN of each LR input patch is adaptively chosen from their associated subset, which significantly improves the speed of synthesizing the HR image while preserving the quality of reconstruction. Experimental results suggest that the proposed method can achieve competitive SR quality compared with other state-of-the-art baselines. Hartigan’s method for k-means clustering is the following greedy heuristic: select a point, and optimally reassign it. This paper develops two other formulations of the heuristic, one leading to a number of consistency properties, the other showing that the data partition is always quite separated from the induced Voronoi partition. A characterization of the volume of this separation is provided. Empirical tests verify not only good optimization performance relative to Lloyd’s method, but also good running time.