In various data settings, it is necessary to compare observations from disparate data sources. We assume the data is in the dissimilarity representation and investigate a joint embedding method that results in a commensurate representation of disparate dissimilarities. We further assume that there are "matched" observations from different conditions which can be considered to be highly similar, for the sake of inference. The joint embedding results in the joint optimization of fidelity (preservation of within-condition dissimilarities) and commensurability (preservation of between-condition dissimilarities between matched observations). We show that the tradeoff between these two criteria can be made explicit using weighted raw stress as the objective function for multidimensional scaling. In our investigations, we use a weight parameter, @math , to control the tradeoff, and choose match detection as the inference task. Our results show weights that are optimal (with respect to the inference task) are different than equal weights for commensurability and fidelity and the proposed weighted embedding scheme provides significant improvements in statistical power.
Our problem is very similar to the 3-way multidimensional scaling problem where the dissimilarity data is a ( @math ) tensor which represent pairwise dissimilarities between @math objects as measured in @math different conditions. However, 3-way MDS methods @cite_2 @cite_0 find a single configuration of @math points representing each object (which is referred to as group space), that is as consistent as possible with the dissimilarity data under different conditions. DISTATIS @cite_12 accomplishes this goal by finding a compromise inner product matrix that is a weighted combination of the inner product matrices in different conditions. 3-way MDS methods such as INDSCAL @cite_2 , PROXSCAL @cite_0 assume a common configuration in group space, from which the individual dissimilarity matrices are computed after being distorted by weight matrices. In contrast, our embedding approach always results in a distinct point for each object under each different condition and we never estimate the representation of the objects in the common space. In fact, some of our inference tasks such as match detection makes sense only if we represent each object under each condition as a distinct point.
@cite_0 @cite_12 @cite_2
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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 ∆. We discuss methodology for multidimensional scaling (MDS) and its implementation in two software systems, GGvis and XGvis. MDS is a visualization technique for proximity data, that is, data in the form of N × N dissimilarity matrices. MDS constructs maps (“configurations,” “embeddings”) in IRk by interpreting the dissimilarities as distances. Two frequent sources of dissimilarities are high-dimensional data and graphs. When the dissimilarities are distances between high-dimensional objects, MDS acts as a (often nonlinear) dimension-reduction technique. When the dissimilarities are shortest-path distances in a graph, MDS acts as a graph layout technique. MDS has found recent attention in machine learning motivated by image databases (“Isomap”). MDS is also of interest in view of the popularity of “kernelizing” approaches inspired by Support Vector Machines (SVMs; “kernel PCA”).This article discusses the following general topics: (1) the stability and multiplicity of MDS solutions; (2) the analysis of struc... Domain specific (dis-)similarity or proximity measures used e.g. in alignment algorithms of sequence data are popular to analyze complicated data objects and to cover domain specific data properties. Without an underlying vector space these data are given as pairwise (dis-)similarities only. The few available methods for such data focus widely on similarities and do not scale to large datasets. Kernel methods are very effective for metric similarity matrices, also at large scale, but costly transformations are necessary starting with non-metric (dis-) similarities. We propose an integrative combination of Nystrom approximation, potential double centering and eigenvalue correction to obtain valid kernel matrices at linear costs in the number of samples. By the proposed approach effective kernel approaches become accessible. Experiments with several larger (dis-)similarity datasets show that the proposed method achieves much better runtime performance than the standard strategy while keeping competitive model accuracy. The main contribution is an efficient and accurate technique, to convert (potentially non-metric) large scale dissimilarity matrices into approximated positive semi-definite kernel matrices at linear costs. HighlightsWe propose a linear time and memory efficient approach for converting low rank dissimilarity matrices to similarity matrices and vice versa.Our approach is applicable for proximities obtained from non-metric proximity measures (indefinite kernels, non-standard dissimilarity measures).The presented approach also comprises a generalization of Landmark MDS - the presented approach is in general more accurate and flexible than Landmark MDS.We provide an alternative derivation of the Nystrom approximation together with a convergence proof, also for indefinite kernels not given in the workshop paper as a core element of the approach. Current implementations of multidimensional scaling (MDS), an approach that attempts to best represent data point similarity in a low-dimensional representation, are not suited for many of today's large-scale datasets. We propose an extension to the spring model approach that allows the user to interactively explore datasets that are far beyond the scale of previous implementations of MDS. We present MDSteer, a steerable MDS computation engine and visualization tool that progressively computes an MDS layout and handles datasets of over one million points. Our technique employs hierarchical data structures and progressive layouts to allow the user to steer the computation of the algorithm to the interesting areas of the dataset. The algorithm iteratively alternates between a layout stage in which a subselection of points are added to the set of active points affected by the MDS iteration, and a binning stage which increases the depth of the bin hierarchy and organizes the currently unplaced points into separate spatial regions. This binning strategy allows the user to select onscreen regions of the layout to focus the MDS computation into the areas of the dataset that are assigned to the selected bins. We show both real and common synthetic benchmark datasets with dimensionalities ranging from 3 to 300 and cardinalities of over one million points